I’d answer “quite well,” but there’s a part one to this ask, and my editor has preemptively sent me a very stern warning punctuated with a screenshot from the latest episode of Binaclebob Trapezoidpants.
That said, the short answer is that it’s very similar to Metronome (whose mechanics I’ve talked about here: http://bills-pokedex.tumblr.com/post/167427470140/how-do-moves-like-metronome-work), just on a limited basis. That is, whereas Metronome may call upon any move in existence, Assist only calls upon moves known by the members of a single party, including the user themselves and party members that have yet to be born (i.e., eggs). How it does this is a bit of a mystery in the pokémon research community, although there are theories, two of which are strongly related to the theories behind Metronome.
The first theory is that all moves are hard-coded into every pokémon’s DNA (as every pokémon is the descendant of Mew, which supposedly could learn a wide variety of moves); it’s just that most techniques have been blocked off by eons of evolution and the development of physical blocks. However, as with Metronome, Assist allows a user to temporarily overcome these blocks and call upon moves that would otherwise be inaccessible to them. The difference is that while Metronome requires no examples to use, according to this theory, Assist actually calls upon memories of the user’s teammates either using these moves or educating the user about how they’re performed. In other words, the skitty (or other Assist user) watches the move being used and emulates it later through Assist. Unfortunately, the problem with this theory is that Assist can also call upon the moves of unborn teammates or moves that a teammate hasn’t yet had the opportunity to use in the first place, but in terms of practicality, this theory makes a bit of sense.
Personally, though, I prefer the second theory, which is that the user themselves don’t call upon the technique Assist emulates but rather the move itself does. However, unlike Metronome, Assist operates on the bonds between the user and their teammates (including the bonds a skitty or what-have-you has with an egg) in order to detect aural potential and project a randomly chosen move. In other words, I feel that there’s a deep connection between a pokémon’s abilities and its life force, that Assist uses a pokémon’s feelings towards others to reach out and detect abilities as encoded in their life force, and finally that Assist uses this information to temporarily bestow upon a user the ability to use one of their close friends’ powers.
In other words, a lot of it is magic, but it’s an explainable magic.
The Pokédex According to Bill 