Why does it have so many followers, and how does it contrast with other versions of Islam?
How has it influenced the politics of Islamic (and non-Islamic) nations?
This question can have different answers depending what one wants to define as being "true" Wahhabism. But since I think it is really mean to ask about the Saudi ideological influence... What happened ideologically in the 1960s and 70s was a "cross-pollination" of (original) Wahhabi and Salfist ideas (of Egyptian origin). This was due to king Faisal letting in and giving support to groups like the Muslim Brotherhood, due to their fight against Nasser:
The Egyptian Muslim Brotherhood represented a source of qualified educators, bureaucrats and engineers, many of them anxious to leave Egypt. During the late 1950s and the 1960s, the Middle East was gripped by a struggle between the traditional monarchies and the secular pan-Arab radicals, led by Nasser’s Egypt, with the pan-Islamist Salafis an important third force. By embracing pan-Islamism, Faisal countered the idea of pan-Arab loyalty centered on Egypt with a larger transnational loyalty centered on Saudi Arabia. During the 1960s, members of the Egyptian Muslim Brotherhood and its offshoots, many of them teachers, were given sanctuary in Saudi Arabia, in a move that undermined Nasser while also relieving the Saudi education crisis.
Saudi Arabia’s foreign policy concerns eased in 1970 with Nasser’s death. But in the 1970s, the Saudi education system was awash with Egyptian Muslim Brothers and other Salafis, much as Berkeley was awash with Marxists. Under King Khaled (r.1975-1982), some of the most important proponents of Qutbist terrorism, including Abdullah Azzam, Omar Abd al-Rahman and Muhammad Qutb, served as academics in the Kingdom. [...]
Although Salafism and Wahhabism began as two distinct movements, Faisal’s embrace of Salafi pan-Islamism resulted in cross-pollination between ibn Abd al-Wahhab’s teachings on tawhid, shirk and bid’a and Salafi interpretations of ahadith (the sayings of Muhammad). Some Salafis nominated ibn Abd al-Wahhab as one of the Salaf (retrospectively bringing Wahhabism into the fold of Salafism), and the Muwahideen [followers of al-Wahhab's original ideas] began calling themselves Salafis.
Today, a profusion of self-proclaimed Salafi groups exist, each accusing the others of deviating from ‘true’ Salafism. Since the 1970s, the Saudis have wisely stopped funding those Salafis that excommunicate nominally Muslim governments (or at least the Saudi government), condemning al-Qaeda as ‘the deviant sect’. The pro-Saudis correctly trace al-Qaeda’s ideological roots to Qutb and al-Banna. Less accurately, they accuse these groups of insidiously ‘entering’ Salafism. In fact, Salafism was imported into Saudi Arabia in its Ikhwani and Qutbist forms. This does not mean that the pro-Saudi Salafis are necessarily benign – for example, Abu Mu’aadh as-Salafee’s main criticism of Qutb and Muslim Brotherhood founder Hassan al-Banna is that they claim Islam teaches tolerance of Jews.
So there's basically a spectrum of Wahhabi-Salafist movements. Those which align closer to the ideology favorable to the Saudi kingdom receive financial support.
And probably the most important example of the spreading of Saudi-falvored Sunni Islam is in Pakistan.
[A leaked] U.S. Consulate cable to the State Department in 2008 [...] goes on to discuss the quantity of the funding and to identify their sources more precisely: “[Pakistani] government and non-governmental sources claimed that financial support estimated at nearly 100 million USD annually was making its way to Deobandi and Ahl-e-Hadith clerics in the region from ‘missionary’ and ‘Islamic charitable’ organizations in Saudi Arabia and the United Arab Emirates ostensibly with the direct support of those governments.”
All this has changed the kind of Islam practiced and taught in Pakistan, which had traditionally had more of a Sufi bent. While the roots of the Deobandi tradition go back to madrassas founded as part of the Islamic revival in 19th-century India, the Deobandis fundamentally oppose Sufi or “folk” Islam and its central concept of intercession by saints (the Barelvi tradition subscribes to this notion). The Ahl-e-Hadith tradition is even more puritanical and is also linked to extremist groups. [...]
By some estimates, 80 percent of the madrassas in Pakistan are now Deobandi. Pakistan’s two main Islamist parties, the Jamiat-Ulema-e-Islam and the Jamaat-e-Islami, follow the Deobandi tradition, which came out of the madrassa at Deoband in Uttar Pradesh, founded in 1866 in colonial India. The Taliban, too, follows an extreme version of the Deobandi faith. Vali Nasr argues that Saudi Arabia is responsible, both intellectually and financially, for the Deobandi resurgence across the Muslim world.
And while Saudi Arabia has had a close relationship with Pakistan’s Deobandi Islamist parties, this alliance has also had its limits. Pakistan’s political Islamists try to not be overtly sectarian, and, according to Nasr, that pushback has led Saudi Arabia to look elsewhere to fulfill its sectarian ambitions in Pakistan—specifically, to the Ahl-eHadith Ulema, thereby empowering them.
In general, the “Saudi-ization” of Pakistan and the changing nature of the country’s religiosity are driven from the ground up—through the influence of madrassas and the Ulema—rather than from the top down [...]
Two historic events in 1979—the Islamic revolution in Iran, and the Soviet invasion of Afghanistan— increased Saudi influence in Pakistan thereafter. The revolution in Iran increased the Saudi imperative to cultivate Pakistan’s Sunnism, and the Afghan jihad and the training of mujahedeen that Pakistan embarked on gave the Saudis a vehicle to do so, via the funding of Deobandi and Ahl-eHadith madrassas in Pakistan. These madrassas served up both the manpower and ideology that fueled the Afghan jihad. Sources suggest that the number of madrassas grew exponentially during this time. According to Mariam Abou-Zahab, “mosques and deeni [religious] madrassas with sectarian affiliations were built everywhere [during this decade], often on state lands.”
This perfectly complemented Zia’s parallel Islamization project in Pakistan, in which the Jamaat-e-Islami was his accomplice as he set out to Islamize public school curricula.
tl;dr: Wahhabism does not have many followers. There are various definitions but it is generally seen by Muslims themselves as a very strict and intolerant version of Sunni Islam. While it played an important role in the history of Saudi Arabia even in recent decades, its influence is now very limited.
The "pivotal idea" of Ibn Abd al-Wahhab's teaching was that people who called themselves Muslims but who participated in alleged innovations were not just misguided or committing a sin, but were "outside the pale of Islam altogether", as were Muslims who disagreed with his definition.
Since the 18th, Wahhabism played an important role in the history of the Kingdom of Saudi Arabia (KSA). This started with the alliance in 1744 between ibn 'Abd al-Wahhab and Muhammad ibn Saud, the founder of the Al Saud dynasty. "The alliance between the Wahhabi mission and Al Saud family has "endured for more than two and half centuries" (quoted from Wikipedia). In a nutshell, Wahhabism provided the Al Saud dynasty with the religious justification of representing the "true Islam".
In the modern history of KSA, there are two major events which shed light on the important but complex role of religious conservationism in the Saudi society:
Because of these two events, the Al Saud royal family took a radically conservative turn to protect the religious credentials of the country and their own as legitimate leaders. More recently, in 2017 the crown prince and de facto ruler Mohammed bin Salman officially recognized this radical turn took 40 years ago and vowed to bring the country back to "moderate Islam".
It is important to emphasize that the vast majority of Saudis don't recognize themselves as followers of Wahhabism nowadays (they simply define themselves as Sunni Muslims). In fact, it is very difficult to precisely define Wahhabism nowadays, as illustrated by the unusual plural in Wikipedia's definitions. But in any case it is incorrect to say that "Wahhabism has many followers". Wikipedia mentions that:
Estimates of the number of adherents to Wahhabism vary, with one source (Mehrdad Izady) giving a figure of fewer than 5 million Wahhabis in the Persian Gulf region (compared to 28.5 million Sunnis and 89 million Shia)
The majority of Sunni and Shia Muslims worldwide disagree with the interpretation of Wahhabism, and many Muslims denounce them as a faction or a "vile sect"
While it is usually considered that Wahhabism has inspired the ISIS extremely intolerant ideology and other forms of religious extremism, its actual political power is very limited: its only potential influential supporter is KSA*. It is commonly accepted that KSA had secretly supported terrorist groups such as Al Qaeda in the past. However for various reasons:
the country seems to be on a path opposite to the Wahhabism doctrine. Of course, KSA is still very conservative by modern standards and is still actively trying to export its conservative vision of Islam, but the recent societal changes brought by Mohammed bin Salman show a direction opposite to the Wahhabism doctrine.
PS: for a detailed account of the historical role of Wahhabism in KSA and its recent downwards trend, see this link.
*: the only other candidate country for religious leadership is Iran, but Iran is 90% Shia Muslim which is completely incompatible with Wahhabism.
Wahhabism is a term which is somewhat ambiguous. Saudis who might be described as Wahhabis do not like the term, as there the word has tribal rather than religious connotations. If we're referring to the puritan Islam of Saudi Arabia, then the word Salafi can be used too, and may be more appropriate.
The movement was born in 1744 when Muahammad ibn Abd al-Wahhab made a pact with tribal leader Muhammad bin Saud. Wahhab's ideas were considered heretical by most back then, as his preoccupation with the threat of 'polytheism' and idolatry (shirk) undermining the oneness of God (tawhid) was regarded as excessive and wrongheaded.
Nonetheless, this did provide a justification to conquer their neighbours, who were suddenly deemed insufficiently Muslim. Consequently by the early 1800s the influence of the Saudi royal family had reached the extent of the modern borders of the Kingdom of Saudi Arabia.
However, there were still issues to be dealt with by the early 1900s, namely the independence of regions like Hejaz. In order to secure control of the Arabian peninsula the Saudi dynasty converted many of the nomadic Bedouin, and in doing so settled them. This produced a surplus of zealots who assembled an army known as the Ikhwan.
The Ikhwan were fundamentalists who fought only with weapons available at the time of Islam's prophet. Regardless, they were effective shock troops, and ensured the political domination of the Arabian peninsula by the House of Saud. Unfortunately for the Saudi family, the Ikhwan were restless, and quickly found issue with what they considered the religious shortcomings of the Saudis themselves. This culminated in the Battle of Sabhilla, where soldiers loyal to the House of Saud slaughtered the Ikhwan with machineguns.
It's quite possible that without the discovery of oil in Arabia, Salafi Islam would have remained culturally isolated, but the exceptional wealth generated by Saudi oil exports allowed the Saudi state to buy influence around the world under the pretence of religious charity. This manifested in 'charitable' activities, such as building mosques and schools... which only taught beliefs aligned with those of Saudi's clerics. This led to the growing influence of Salafi beliefs in places like Pakistan.
As the behaviour of the Ikhwan may have suggested... Salafism is an unstable ideology which is prone to violence. In 1979 militants believing their leader to be the Mahdi (an apocalypse-inducing character who shows up a few years prior to the Day of Judgement) seized the Grand Mosque in Mecca.
Saudi authorities prevailed after a couple of hundred deaths, but king Khalid's response was to compromise with exceptionally intolerant Salafi reforms. It has been said that this is because the king was mindful that his predecessor, king Faisal, had been assassinated a few years prior in 1975 for sweeping modernisation, including the introduction of television; which provoked violent protests.
The rise of Salafi beliefs in the Middle East can be understood as a consequence of the failure of secular ideologies to achieve pan-arabism. In 1966 Sayyid Qutb was executed in Egypt. Qutb is credited as being the first modern Islamist. At the time it didn't look like theocratic beliefs would dominate the region. Palestinian liberation movements were secular, and Arab republics were supported by the Soviet Union. Arab women were influenced by relatively feminist ideologies: American liberalism, Soviet socialism, Turkish Kemalism.
In 1981 Egyptian President Anwar Sadat was assassinated by members of Egyptian Islamic Jihad. By this point many Arabs regarded the region's governing presidents and kings as corrupt sellouts, and with the collapse of Communism in eastern Europe in 1989 the only ideology left which provided a narrative for national liberation was religious. After the Iranian Revolution (1979) theocratic organisations emerged to challenge secular power. Notably: Hezbollah (1985), Palestinian Islamic Jihad (1987), Hamas (1987).
Combining an ideological vacuum in the Middle East, with a lingering desire for national liberation, Saudi oil money, revolutionary Islamism, and widespread poverty, it's easy to understand how Salafi beliefs spread amongst Sunni Arabs.