Pornography and (online and on-site) gambling are illegal in South Korea. The country is very conservative and it takes what it considers to be issues very seriously. For example, South Korean citizens cannot even visit casinos abroad and citizens that are guilty of gambling abroad may be prosecuted when they return to their country.
The ban on pornography is partly motivated by child protection. In addition, crimes against women (physical violence, rapes, revenge porn,...) have been a hot issue in the recent years.
From the Korean Times,
Meanwhile, women's rights organizations have praised the KCSC for taking what they say is necessary action to protect women. They said allowing access to porn sites, where revenge porn is not just found but plays a major role in attracting people, is tantamount to turning a blind eye to victims' suffering.
Another reason for the censorship is that a large part of the population is still very conservative and religious groups, in particular Christian churches, have a lot of influence. Watching pornography and gambling are considered by these groups as immoral. From the same article of the Korean Times,
The Korea Communications Standards Commission (KCSC), an internet censorship body, said on Feb. 11 that it had blocked access to 895 overseas-based websites with "harmful" content, including Pornhub, the world's largest porn site.
According to Koreabizwire,
From a legal standpoint, the term ‘pornography’ is not clearly defined. Some precedents make it more ambiguous when they call pornography “placed at the borders of obscenity and art”.
The term ‘obscenity’ also varies in definition among a number of legal precedents.
While one case would define obscenity as a stimulant for sexual desires that violates morality, another case would say that obscenity can be judged based on conventional understanding.
South Koreans are subject to heavy punishment for taking or distributing pornographic videos. The online distribution of pornography can be punished by up to 1 year of imprisonment or a fine of 10 million won (US$8,921).
I couldn't find any statement on the new policy from the government but the previous censorship methods were notoriously inefficient and considered as a joke by many South Koreans. Basically, most websites offering pornography had an https version that was not blocked before. Source: I am currently living in South Korea.
Also for the Korean Times article,
Simply put, if anyone tries to access a harmful website using https, the government opens up the data packets to find out the web server the person wants to access. While other information in the data packet is encrypted, the server address is not. Therefore, the government can block such access.
We can guess that the South Korean government considered this is an acceptable sacrifice of freedom and privacy in order to assure the safety of its citizens. In recent years, this is not an unprecedented action from a democratic country (think NSA, or the Emergency State in France after the terror attacks).