The story
Unknown. First appeared in print in 1980s, but probably existed before that. The most common theory for its origin is that a purchase (especially that of a full three-piece suit) from Montague Maurice Burton (1885–1952), founder of Burton Menswear, was known as a "full Monty". According to the OED, this etymology is "perhaps the most plausible". The nudity definition comes via the movie The Full Monty, in which it is used as a euphemism for removing all one's clothes when stripping.
Etymology from Wiktionary, CC BY-SA 4.0.