If you read the book, the character of Tom isn't subservient to whites or a "sell-out," yet his name has become an accepted pejorative for such well beyond the US.
Back in the 19th century, intellectual property law was not nearly as robust as it is today. Uncle Tom's Cabin, being a very popular book, inspired many, many derivative works and parodies. None of them were necessarily individually as popular as the original, but taken together they were inescapable parts of the culture.
Minstrel shows in particular latched onto the Uncle Tom character, and slowly turned the character into an old, impotent buffoon who kowtowed to his white masters. This "Uncle Tom" character, derived but very far away from the character in the book, was what inspired the epithet.