The general consensus I’ve read is that George I’s reliance on his ministers - he was originally from Hanover and did not speak English - lead to the development of the quasi-modern form of parliamentary government that evolved into what we see today. Robert Walpole is generally considered the first Prime Minister as a result of this.

In the century before, conflicts between the powers of the King and Parliament led to the English Civil War and the execution of Charles I. After the Restoration and the Glorious Revolution, Parliamentary authority has been somewhat asserted by their declaration of Charles II and later William and Mary as monarchs. When George I ascended to the throne, he relied heavily on his ministers, due to the aforementioned conditions, and this tradition kept. Over time, the voting process was reformed as the franchise was expanded and districts were reorganized, and the powers of the monarch were gradually limited more and more. There isn’t necessarily a direct point where the transition happened, but the origin of the modern Parliamentary system could be said to have begun with the reign of George I.

In case there are no other direct responses to your query, you'll find some quite extensive discussions of what is essentially the same question on these threads:

Which British monarch was the last one to have "real power?"

A slightly different, legalistic, point of view is provided by

How and when did the British monarchs lose the rest of their executive and legislative power?