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.
In addition to @Glorfindel -part of the answer is also that top players really are extremely fast. They recognizes the position and patterns faster than us mortal can blink. Some of the fastest chess is played with 15 seconds on the clock. That means that the whole match is over in less than half a minute.
A search for Ultrabullet chess returns e.g. https://www.youtube.com/watch?v=PLVK7iryHJA
or here both players berserks (meaning they only have half the time rounded e.i. 8 seconds) and play blindfolded: https://youtu.be/6ygQMw4rBHg?t=10m54s
Here: https://youtu.be/G2AMN9tHSB0?t=1h1m25s 87 moves was made in 13.5 second. That is more than 6 moves/second.
When the chess is so fast it is difficult to tell if the clock had started counting when the move was made or if it was a premove made when the opponents clock was counting. Often you will hear players complain about lag, as even few ms of internet/server delay affects their play.
When you see moves happening in milliseconds, this is usually do to premoves. Players pre-select the move they want before it's there turn, and the server does their move immediately. I believe it used to take 0.1 seconds before, but now it might be even less time (I'm not too sure).
However, I've often played out an opening in only 2-3 seconds without premoves. If you know the theory, or it's just a middlegame position that doesn't require any calculation, moves can be played very quickly. Basically muscle memory and instinct. It also helps to hover the piece over the square you want to move it before your opponent makes their move, so all you have to do is drop the piece. Of course, if your opponent is also moving at the same speeds you are, this isn't always possible.
Correctly made premoves save a lot of time. In addition, best bullet players just have a better neurological makeup - meaning faster decision-making and reflexes - than us, mere mortals, do. You also need to have a good mouse and fast internet. For some insights of what it takes to win in blitz and bullet at the high level just watch, for example, Nakamura's streams on Youtube.
