The game is justifiably famous and is linked here.
[FEN ""] [startply "39"] 1.e4 c6 2.d4 d5 3.Nc3 dxe4 4.Nxe4 Nd7 5.Ng5 Ngf6 6.Bd3 e6 7.N1f3 h6 8.Nxe6 Qe7 9.O-O fxe6 10.Bg6+ Kd8 11.Bf4 b5 12.a4 Bb7 13.Re1 Nd5 14.Bg3 Kc8 15.axb5 cxb5 16.Qd3 Bc6 17.Bf5 exf5 18.Rxe7 Bxe7 19.c4 1-0
It was a shocking upset at the time because
19. c4 looks completely harmless in this position.
Analyses after the match and around the time predicted the position would be roughly even for several iterations; Kasparov could have held out. Kasparov had battled Deep Blue at least eleven times prior to this match (six times in 1996 and five uptil then in 1997) and had managed to win convincingly against it, albeit with some losses - but this had not deterred him before.
Which is why it is all the more surprising Kasparov gave up at this exact point in the game.
What was going on in Kasparov's head that made him decide he could not possibly continue after c4? Was it simply the idea that Deep Blue must be cheating? Was it some set of lines that made him quail? Were other events around the day eating at him?
I'm completely mystified by this decision, and would love to have some answers!