tessathedog: I think john2629 has grasped it. Karpov wins by grabbing space on the kingside with his pawns. But there's much more to this wonderful game than that. There are several highly instructive moves, both tactical and strategical. For example:
15...b6! Not castling just yet, because with the black pawn on h6, immediate castling invites 16 Be3 and 0-0-0 followed by a kingside pawn storm. Note how Karpov DOES castle kingside next move after White has committed his king to that side of the board.
22...Nf6! This seems to invite 23 Ne5, but that's refuted tactically with 23...Rd5! 24 Rfe1 Bxf2+!! (lovely move. The Queen can't take on f2 because she is overloaded protecting the Knight on e5, and taking with the King runs into ...Rxd5 and...Ng4 forking the King and Queen). Beliavsky must have seen this and so plays the otherwise hard to understand backwards move 23 Nd2 instead.)
It's one thing to gain space, and another thing to know how to use it. Karpov's mastery of how to use it is revealed in the variation 43 Rh1 (preparing the simplifying h3) Rxh4 44 Bb5 (the bishop has to move before h3 can be played, but this tempo is crucial) 44...f5! 45 h3 f4! 46 hxg4 f3 47 Kg1 Rxg4+ with advantage.
Finally, 48...g3! to clear the path for the "e" pawn. Clearly foreseen the move before when 47...e4 was selected.
All of these moves are of the type that separate a super GM from a regular one. One can understand and appreciate them once they are shown to you (Karpov's own notes revealed them to me), but it takes a special insight to find them over the board.