According to the dictionary of slang:

One thing can be said about this once fairly common used British slang phrase... that it now has global familiarity within the English speaking world thanks to the acclaimed 1997 film of the same name. Few people ever expected this traditional and typical British film comedy to ever make the big time globally, but once it hit the shores of the States there was no looking back for the phrase. As the popularity of the movie spiralled, interest in this unusual expression increased and the UK lost another piece of its enigmatic vernacular to the world. For those who still don't know the real meaning of the phrase 'the full monty', it means the whole amount. Unfortunately its true origins, as with much slang, can only be speculated upon; what is known is that although previously used mainly in the North of England it really gained popularity throughout the UK in the early 1990's. After the film, the term has come to mean ‘going all the way’. While the show works toward ‘going all the way’, the real question is do the actors on stage actually go ‘the full monty’? You’ll have to see the show to find out.