To add to Gerry's suggestion, I remember hearing a comment of the composer Ravel: he said "Copy. If you have some originality, then this may come out as you copy. If not, then never mind." I would add that the originality may come out only after you copied several, or many times. I have also heard it said that Newton was an inveterate copier.
So rather than just read proofs, I would suggest you copy proofs by hand, and this may get you gradually into the rhythm of them. Also, you can ask: "What is the key idea?"
In constructing proofs, you also have to learn to work from both ends, forward from the assumptions, and backwards from the conclusion, and hope they meet! Proofs are rarely constructed linearly, you need to know where you are going.
In teaching analysis, I also used the "fill-in" kind of exercise. Take a complicated proof, such as the product of limits is the limit of the product, write it out, then blank out bits, and ask the students to fill in the blanks, giving lots of clues from the parts that are still there. Thus the whole structure of the proof is there, but the students have to use the clues from what is there to complete the details. An advantage is that solutions are easy to mark!
The teaching method behind this is known as "reverse chaining" or "backward chaining", see wikipedia, for example. In other words, the first task you give is to complete something easy. Then you gradually make it more difficult, or start at an earlier stage. The method is a standard training technique, for say animal training, or teaching a child to put on its clothes. The child learns from success, so you arrange for it to be successful! Learning mathematics at any level has some of the same character. It is also how you learn to do many puzzles such as Sudoku, or crosswords, start with the easy ones.