My capacity to read proofs went through the roof once I went through a few chapters in Velleman's How to Prove It: A Structured Approach. If you feel shaky with less intuitive proof techniques like proof by contrapositive or , for example, how to prove a logical disjunction of propositions, you should at least skim parts of this (or a similar) book.

Echoing what other posters' have said: always read with a pencil and notebook in front of you and write down all the definitions you read. Draw pictures. In my experience, a large part of being able to comprehend more advanced mathematics is being able to take complicated definitions and chunk them into a single concepts without losing the fine details of the definition. I've found that asking yourself dumb questions about definitions is useful for trying to do this.