Deine Geschichte nehme ich dir nicht ab.

English Translation

I’m not buying your story.

I don’t get how “abnehmen” is being used here to mean “believe” or “buy”. Is this slang?

This is a fixed expression , often shorter
“Das nehme ich dir nicht ab” = Das glaube ich dir nicht"
It’s colloquial but not slang.

It is noteworthy that the saying does not exist without the negation.
Das nehme ich dir ab " would be very unusual.