How do sleeping Pokémon even escape their Pokéballs?

Question

They're SLEEPING! How do they even break free? Don't give me the typical 'Because Game Freak made/wanted it that way.)

Answer 1

^ That is the formula for the modified capture rate. As you can see on the last part you times the catch rate by any bonus status conditions the target Pokémon has, if any. If the target is asleep, then it is 2x.
So Sleep does affect the probability of a Pokémon breaking out, but it doesn't guarantee a capture, that would be too easy, and the game doesn't need to use logic.

Answer 2

If you're asking about how an already sleep Pokemon comes out of its ball into a battle, then it's pretty evident that the Pokeball is controlled by the trainer, not the Pokemon, which means that it's the trainer's decision to call a Pokemon into a battle, regardless of the Pokemon being asleep.

Comments

If someone grabs you in your sleep don't you pull away?
When a Pokemon is pulled into the pokeball it feels the change and struggles in it's sleep. However its still only half as effective as when it's awake.