How do top players move so fast? I've seen 3 minute and 1 minute matches on Chess.com where the players move at insane speeds during the last few remaining seconds of the match. 

I'm talking in milliseconds, I'd at least take that much of time to only recognize an event happening. So how do they even respond with a correct move?

How do they move so fast? Do they use some kind of special software or what?
You're likely looking at premoves; the players know (or at least have a good guess at) what their opponent's move is going to be, so they can already 'ready' their own move by taking the piece and dropping or just moving it to the destination square. Of course, if the opponent makes another move, they have to cancel their move (if it's still possible; that depends on the client, and whether they're dropping or just moving) or they might make a huge blunder.

Premoving can lead to funny things like the following:

[FEN ""]
1. d4 d5 2. Bg5 e6 3. Bxd8


Here, Black expects White to play a Queens Gambit with 2. c4, which is the normal move. If Black premoves 2... e6 and notices the 2. Bg5 too late, they will lose their queen.