Probably the first of only two occurrences where Kasparov's ego-based self-belief collapsed. That mysterious rook move in a drawn endgame spooked him, and seeded the doubt.
Doubt is a costly thing in tournament conditions, doubt about the opponent leads to doubt in your own ability to predict your opponent's moves. So you start looking at more candidate moves at each branch of analysis, and also you start rechecking lines again -- all against the candidate move approach. All this costs time and effort, and tires you down quicker. So weaker patches of play become more regular, unless you buckle down and keep it under control, another source of energy leak.
And over a series of games, in trying to hold that back, Kasparov probably eased off a little in the last game, and committed perhaps the worst blunder of his career, in a line he knew very very well from the other side of the board.
The second time Kasparov's self-belief collapsed was the Braingames World Championship match with Kramnik. But this was less to do with being spooked, and more with the evidence his opponent was far more prepared - perhaps for the first time in Kasparov's career.
Kasparov wears his emotions on his sleeve. A lot of his incredible conceptions are a display of his aggressive creativity and stamina. An emotional battle is one core part of Kasparov's armoury, along with well-prepared innovations and surprises, and his capacity to find aggressive moves that keep his opponent off balance.
I am so impressed for how long Kasparov has kept his game together. Right up to his last tournament in 2006, 25 years playing chess of such quality and magnificence. When others junior to him faded and burned out around him - Shirov, and perhaps recently Morozevic.
And Kasparov, like all chess players, needed to find reasons, external reasons, to explain the failure. In the Deeper Blue match, the paranoia of not seeing analysis printouts from the machine prompted the suggestion that the IBM team were hiding something, like a human grandmaster making that move. It's necessary in the make-up of a chess player to find an external reason, because otherwise it is a clear demonstration of fallibility in tense situations.
(The book "The Inner Game" about Nigel Short's path to challenge Garry Kasparov in 1993, talks a bit about how chess players need to put disasters out of their mind, to be able to play the next game in the match. That ability allowed Short to come out in the next game and sometime produce amazing moves and pushed Kasparov, despite the heavy losses.)
The interesting part of the Deeper Blue match isn't that the computer won the last game, but the human factors of that mysterious rook move right at the start of the match. It set off an emotional reaction in Kasparov, which lead to the spectacular collapse in the last game.
Interesting too is how Kasparov somehow overcame this collapse. Somehow the effects of this disaster didn't seem to affect the rest of his playing strength. He somehow regained complete confidence in himself.
I don't think his match loss to Kramnik was affected by this, Kasparov was just outthought, outplayed, and out prepared by a better player. Kramnik rose to a fantastic level to win that match. Kramnik's fitness for one, was impressive considering his past lifestyle.