It mostly just takes practice. A good way to get into it is with property-based testing. It's less formal, but you end up expressing many of the same things (you're at least expressing the post-conditions, if not the rest of the things needed for a proof). With practice, you'll start to understand your systems better, and things like what I wrote up will come to you more easily when analyzing and designing them.

To move towards formal proofs of code, I like Leino's Program Proofs (uses Dafny), one of the more approachable tutorials on the subject.