It's just an analogy, you're not supposed to think too deeply about them.

The main take away for a lay person is that _like_ the mattress space is being deformed. That's where the analogy stops. Taking it further, like with all analogies, breaks the analogy.

If the analogy was a perfect one, then it would just be the reason and not an analogy.

My main gripe is how hard for most people it is to extrapolate that deformed mattress into a 3d space.