I think it's very outdated. I know it from books but I've never heard it used in real life. (I'm a U.S. native speaker). It means "I don't believe it." The implied context is an imaginary bet. "I'm so sure it is untrue, that I'm willing to bet on it. If I lose, I have to eat my hat." Some other phrases that mean "impossible:" "... then I'll be a monkey's uncle." (Outdated). "...then I'm the Queen of Sheba." (Possibly regional or ethic? The "impossibility" aspect is that "the Queen of Sheba" = the ultimate in beauty.) "...when pigs fly." "...when hell freezes over." None of these are "bad language" but they are basically impolite because they express an EXTREME degree of disbelief. Steve Jobs had said that Apple would never create any Windows software. In 2003, he introduced his announcement of iTunes for Windows with a big slide that said "Hell froze over."

This expression basically means to be in disbelief about an event occurring. "There is no way he did that, but if he did I'll eat my hat." Hopefully I helped, this expression isn't as common in the United States.

I wouldn't say it's common but your hear it from time to time in the UK. Recently a politician, Paddy Ashdown said he would eat his hat if the exit polls of the general election turned out to be right. He of course only said it because he was very confident that they would be wrong. Well it turned out he was wrong and he had to eat his hat on national TV. The only difference was they gave him a chocolate cake in the shape of a hat, rather than an actual hat to make it edible lol.