When did the British monarchy actually hand over its legislative powers to the parliament? Why is the monarchy still present today? There seems to be no real need for it anymore. Not only that I think it costs British people much more to have a sovereign than having a head of state.
The modern day sovereign of England realistically has no powers in government today. They are a figurehead kept around for tradition and historical reasons. The sovereign does actually still have official powers, but the sovereign only uses them at the behest of the elected government, namely the prime minister.
There is a nice writeup of the Queen's current power on the monarchy's website. The powers that relate to legislation are:
Summoning/Proroguing Parliament – The Queen has the power to prorogue (suspend) and to summon (call back) Parliament – prorogation typically happens at the end of a parliamentary session, and the summoning occurs shortly after, when The Queen attends the State Opening of Parliament.
Royal Assent – It is The Queen’s right and responsibility to grant assent to bills from Parliament, signing them into law. Whilst, in theory, she could decide to refuse assent, the last Monarch to do this was Queen Anne in 1708.
Secondary Legislation – The Queen can create Orders-in-Council and Letters Patent, that regulate parts to do with the Crown, such as precedence, titles. Orders in Council are often used by Ministers nowadays to bring Acts of Parliament into law.
Appointing the Prime Minister – The Queen is responsible for appointing the Prime Minister after a general election or a resignation, in a General Election The Queen will appoint the candidate who is likely to have the most support of the House of Commons. In the event of a resignation, The Queen listens to advice on who should be appointed as their successor.
Declaration of War – The Sovereign retains the power to declare war against other nations, though in practice this is done by the Prime Minister and Parliament of the day.
Remember though, all these powers are on paper only. In reality all those powers lie with the Prime Minister and Parliament.
The monarchy lost its legislative (and executive powers) over a very long period of time. The starting point would be the Magna Carta (1215), which significantly reduced royal authority. The ultimate culmination would be the English civil war (1642-1651, though there were some shaping events after the war), which effectively ended the monarchy as a body with any power.
So why does England maintain a monarchy despite having hundreds of years of history of limiting the monarchy? Simply put, because it's cool. Not many countries have such rich, long-lasting histories of continuous royalty. It is literally part of the national self-identity.
Beyond the non-tangible effects of the monarchy, there are some very clear advantages to having the monarchy. It is a huge tourism driver. A 2010 telegraph article claimed that the monarchy attracted $500 million pounds per year in overseas tourism. I would expect it's only gone up since then. On the flip side, the Monarchy's income is determined by the Sovereign Grant Act, which states that for 2016-2017 the monarchy only recieved £42.8 million. In 2010 it was £31 million. Thats over a 16 times increase. Sounds like a pretty good reason to keep them around to me.