This quote was derived from Warhol who said it a bit differently , "In the future everybody will be world famous for fifteen minutes".
'15 minutes of fame' is an expression generally used in reference to celebrities short-lived fame. Easy phrase to use to explain the reality TV culture which is rife these days.
Interesting quote, Andy. If I could choose why I would have fame for fifteen minutes it would be for something positive which changes people’s lives in some way, not very original but the best thing to be famous for, I believe.