When you see a proof in print it’s like an Old Master painting – everything in its place and polished and perfect. You get the feeling that no human being could produce such a thing, and you despair. This is why Old Master drawings are so encouraging. Repeated attempts at getting the same thing. There is a lovely Guercino drawing where the hero has three left arms: one because Guercino got the proportions wrong, one because he changed his mind about where to put it.
The description of how something comes about is inevitably far longer than the act of doing it. Try describing a BJJ move in terms of what happens, why it happens, and what would have happened if you’d done something else. Try describing walking, for that matter.
So this description of how to prove the Fundamental Theorem of Arithmetic (that every number has a unique factorization into prime numbers) is invaluable. It describes how one actually thinks when proving things, not merely the result of doing the thinking.