The erosion of the Crown's power was a gradual process. It did start with a bang; the English Civil War of 1642-1648. Previous to the war, Charles I was trying to rule as an absolutist monarch, but he found he couldn't extract enough revenue using the Crown's traditional privileges to run the country. His task was complicated by a Scottish rising in 1639 against his Kirk policy, and an Irish rebellion in 1641 against the Ulster Plantations. He needed to fund two armies, so he called Parliament to raise revenue. Things got out of hand, he raised troops to suppress Parliament, and lost his war, his throne and his head. Parliamentary Supremacy Firmly Established.
The Crown could no longer claim absolute power, but maintained lots of influence through traditional privilege, patronage, Irish revenues, and various odds and ends. The unseating of Charles I's Catholic son, James II, in the Glorious Revolution of 1688, however, confirmed the verdict of the Civil War; the Crown was subject to the law, not the source of law. The Hanoverian succession of 1715, coupled with the "1715" Jacobite rising (in the name of James II's son, the Old Pretender), pushed much of the Crown's remaining privileges into the hands of the leaders of Parliament. The process was aided by the fact that the monarch who took the throne from James II was William of Orange, Stadholder of Holland, who spent most of his time managing his Dutch affairs and leading armies unsuccessfully on the Continent against the French troops of Louis XIV. William's first successor was Queen Anne, who was more hands-on, but not very sensible. Anne was succeeded by the first two Hanoverian monarchs (George I and George II). They both were German by language, culture and inclination; they left much of the day-to-day running of the Royal Establishment in the hands of Prime Ministers like Walpole, Peel, and the like. England became more and more accustomed to having all major government decisions made by the leaders of Parliament, rather than by the monarchs.
It wasn't until George III assumed the throne (in 1757?) that England finally received a monarch native in culture and engaged in the affairs of the nation. George III had a good start with British victories in the Seven Years War. But success went to his head, and his bull-headedness with his American colonials sparked the disastrous War of American Independence. The subsequent British defeat drove George III insane. His wastrel son, George IV, assumed the Regency, but again left the realm's affairs in the hands of Prime Ministers such as the William Pitts (Elder & Younger).
George IV had no legitimate children; his younger brother William IV took the throne. William also died without issue, so their neice Victoria became Queen.
By now, we are 200 years after the English Civil War. In those 200 years, decent monarchs were few and far between; only Charles II and William III even come close, and each had his own peculiar handicaps (Charles II was secretly Catholic, and William III more Dutch than English). The Prime Ministership, on the other hand, had seen such outstanding personalities as Peel, Walpole and the Pitts (not to mention the Duke of Wellington). Under Victoria, the run of fabulous PM's continued, with Palmerston, Disraeli and Gladstone. Victoria followed tradition by deferring to her PM's. She would have assumed direct rule only if her husband, Prince Albert, had insisted, but Albert saw the virtues of a Constitutional Monarchy and encouraged her trust in the existing system. His early death prompted Victoria to remove herself from public life, again serving to enshrine the PMs' authority.
The decline in the influence of the Throne was a gradual process, prompted as much by circumstance as by the wisdom of Britain's governing elite. Given the fate of Europe's other great monarchies (Habsburg, Hohenzollern, Romanov, and Bourbon), the British Royal Family likely considers itself lucky to retain the station it occupies today.