I never understood exactly what TASing is, or what makes up the "tool-assisted" part of it (Does it involves cheats or something? Or is it just using an emulator for more flexibility?), but it's amazing.
Emulators are used for various reasons.
1 - To have frame-by-frame control. Even the fastest IRL player is going to have extreme issues pressing a certain button sequence if each button press and release needs to be a single frame apart for a string of 20 presses. For example building up enough speed with the backwards long jump on stairs in order to push yourself through walls (as seen in that video). In other games the random number generator may depend on when certain buttons are pressed, and you can't maintain frame-precise control of that with human hands on a controller.
2 - For rewinding. The runs are not played in one go, in fact the "rerecord" count (the number of times time was rewinded to try a specific part again) for many runs can be
really high. Hell, for that mario video, the count is 225,577. So as you might guess, a single TAS run can take a long, long time. It's not unheard of at all for a TAS run of a game to take a year or more.
3 - In order to see what the game's actually doing... for example in order to keep track of certain hidden values you may be manipulating that you might not have a counter for normally. You know, stuff like seeing how your actions are affecting the random number generator to try to get a monster to drop a certain item when you kill it, opposed to just running in and killing the damned thing 500 times like a normal player would have to.
An important thing is that cheats are not used, and only specifically-approved emulators (even specific version numbers of those) can be used, so the end result of the TAS is ACTUALLY POSSIBLE in the game (the first goal of a TAS is to beat a game as fast as is possible, period*)... assuming you could get somebody to input that specific button sequence on the controller with enough accuracy.
And actually, for NES games there's some guy that hooked up a machine to his NES's control port or something so he can watch the machine put in the sequences to TAS an NES game, both for fun, and to prove the stuff's true.
*- This doesn't always make for an entertaining run, so many games that can use glitches to skip the whole game (like Super Mario 64) often have additional "categories" for TASes that complete certain goals or try to play through the whole game, so there's also TAS runs of that game for entertainment purposes.
