In short, the King used his political muscle because he could, and because he wanted to. That and a bit of luck.
Things weren't quite so 'constitutional' then.
Remember that the "British Constitution" isn't a single document like the United States Constitution, but instead a grab bag of hidebound habits, and a series of royal and parliamentary ordinances, passed over time to eventually codify the habits.
The Monarch's primary power at the time was the control of patronage; that is, he was able to award offices, knighthoods, and peerages, and this was the typical lubrication used to get someone to vote for something back then. If the Prime Minister asked His Majesty nicely, His Majesty would grant someone a favor that would swing that someone to vote the way the Prime Minister wanted.
In the 18th and 19th centuries, The House of Lords could block anything the House of Commons passed; the only way to break a deadlock was for the Monarch to create peers to tip the balance. If he wanted to.
This actually happened in 1832 when William IV helped Earl Grey pass the Reform Act, finally doing away with rotten boroughs and other such corruptions of the democratic ideal.
It took the 1911 Parliament Act to finally rein in the House of Lords.
And vice versa. Whig politician Charles Fox was a firebrand reformer, denouncing the King's use of political muscle as creeping tyranny.
Fox opposed Tory PM Lord North all through the latter's tenure during the American Revolution. Although the Pitts had both argued for the things the American Colonies wanted, to keep them in the Empire, Fox took it a step further, wearing the colors of the Continental uniform to provoke the Tories and George.
Fox isolated himself politically and had to ally himself with a former enemy on the outs.
Lord North lost control of Parliament when Britain lost the Thirteen Colonies to revolution. But Fox refused to serve under North's Whig successor, Lord Shelburne because of a political slight the latter did to Fox's father back in 1763.
Fox and Lord North formed a coalition government without the King's requesting it.
At any rate, Lord Shelburne didn't live very long, and the two erstwhile enemies were able to get enough allies together to make a government under the Duke of Portland.
The Fox-North Coalition was the first time a government was formed without the Monarch's request to form.
So George was not about to assist Fox. Fox tried to reform the East India Company, and George bullied the House of Lords into blocking the bill, which gave him a pretext to dismiss the Portland (Fox/North) government, and appoint 24-year-old William Pitt the Younger pending an election.
So Pitt was now Prime Minister and had the King's support and patronage. Fox had lost some popularity among reformers due to his alliance with Lord North, but he was still able to engineer a vote of no confidence. Pitt refused to resign.
And then one day, while returning from a ceremony granting him the Freedom of London, Pitt's coach was attacked while passing a Whig Club. People assumed that Fox had engineered it. Pitt's popularity caused candidates in the ensuing 1784 election to think twice about opposing him, and with George III's patronage, Pitt won.