How does Crouch jump actually work?

I'm wondering, in all valve games, you can "crouch jump". The action of crouching just before a jump makes you somehow jump higher.

I get that it's kinda more realistic, but was it an intended (programmed) thing, or is it a wonky hitbox when crouching that permits that effect?

Also, side question: How much more (in % if we count the base jump as reference) do you jump with a crouch jump?

It's definitely an intentional thing. It's featured quite prominently in Half-Life 1's Hazard Course, which is the part of the game that teaches you how to play it.

To my knowledge, it exists in all GoldSrc and Source games to date that aren't top-down (Alien Swarm, DOTA2).

The reason it works is that it more-or-less pulls your model's hitbox up as if you were bending your legs during a jump.

The actual numbers are going to be different for each game, though.