Short version, with old-fashioned confidence-man-style scamming. He did it so effectively that when time ran out on one scam, he just pivoted to the next one, as he was constantly cultivating men and women who became extraordinarily willing to help him. He had the rich-people-manipulation skills of a young Anna Delvey/Sorokin, and boy, did he use them! As the NYT summarises:

He started small, testing his tactics and seeing what he could get away with. His early successes laid the foundation for more ambitious ploys down the road. Again and again, he proved willing to operate on the edge of criminality and burn bridges in his pursuit of wealth and power.

...This is weird: the father of the lady Junior just got engaged to, Bettina Anderson, shows up at the end. Harry Loy Anderson Jr. was the president of the Palm Beach National Bank & Trust Company, where Epstein had held accounts since the early 1990s. In 1999, when Epstein was trying to use his US Virgin Islands property to get millions in tax breaks for his main company, Financial Trust, Anderson wrote a letter to the USVI’s Industrial Development Commission, vouching for Epstein’s character, that he was “a gentleman of the highest integrity,” and that he “enjoys an excellent reputation in our community.” Later, Epstein’s house manager said in a deposition that he was using Anderson’s bank to pay some of Epstein’s victims, too.
...
Epstein got his job as a Bear Stearns trader, it seems, from the parent of one of his students at the Dalton School, who noticed that Epstein was good at math, suggested he get a job on Wall Street, and called up a buddy. Good timing! Epstein had just been asked to leave at the end of the school year, and his flirting with female students had not gone unnoticed.

And this was no ordinary pal! It was Bear Stearns executive Alan Courtney “Ace” Greenberg. Epstein had zero experience in business or finance, and not even an idea how the stock market worked. But Greenberg was into “hiring what he called people with PSD degrees — non-MBAs who were poor, smart and desirous of riches.” And Epstein was that! (Detail from Greenberg’s obituary: “In 1998, he gave $1 million to the Hospital for Special Surgery in Manhattan to provide Viagra to men who could not afford it.” A wangdonger is a terrible thing to waste!)

https://www.wonkette.com/p/oh-look-the-nyt-solved-the-mystery