This week I read a paper comparing how undergraduates and doctoral students/professors address proofs. The chief difference between the two groups was that undergraduates always started with definition and tried to build from there, while the more experienced mathematicians started by examining their intuition about the concepts and then built a proof. While the paper's authors didn't say as much, I took these conclusions to mean that gaining experience with concepts is more important than memorizing precise definitions.
Armed with my new perspective, I took to some arithmetic proofs. I tried to visualize the fundamental theorem of arithmetic, the linear combination form of the greatest common divisor, and so on, but at best I could only put together sketches of proof. What I said looked nothing like the odd answers at the back of the book. I became increasingly frustrated.
This went on for about three days. Finally, I gave up and went off to weld pieces of metal together. After my mind was good and off the subject of arithmetic, I started thinking about proof from the perspective of idea shapes that fit together or compliment each other. The process of imagining math in a way that currently has absolutely no practicable application flicked on the light bulb. I have an intuition for the concepts to be proven, but no intuition for how those concepts interface with the language that is used to convince others. This revelation doesn't tell me how to acquire this intuition, but it is a piece of the puzzle.