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This was nice, but it still wasn't sufficient: I understood the theorems, but they seemed like a random walk through theorem-space. I couldn't yet understand why anyone would say those particular things on purpose.

This is exactly what I was thinking lately and which possibly took a lot of time for me to come to terms.
I initially questioned "why" for every step in a proof and without ever completing the proof, it didn't make sense at all. They seemed random! 
Very recently I've understood that understanding is not continuous but discrete.
I've practiced to hold all of my "why" questions till the proof is completly covered, all suddently everything will fall into the picture. 
Though this works, but is especially harder when for example, we need to prove multiple lemmas to prove a proposition. 
Anyways, I believe this is how it works :)

Btw, I'm coming back to this post from time to time. Everytime I come here, I feel a breeze of a rattle from someone who's went through a similar struggle as of me. It feels good, reminding that I'm not alone.
Thanks for expressing yourself!!