I was in similar situation than yours during my PhD: my proofs were correct, but tedious to read, and every time one my co-author rewrote them, they looked so much clearer! I wouldn't say that now my proofs are perfect, but they have definitely improved. I would associate this improvement with the following factors:
- my english has improved with time, and I can now use more variations of the "proof vocabulary".
- I have been working with different authors, so I've been exposed to different proof styles.
- I have been writing more proofs, and as Artem's perfectly said, the more you write, the more you get feedback, the better you get.
- I have read and given feedback to other people proofs, so I've started to noticed what I like/don't like on a proof that I haven't written.
So I know it sounds like a dull advice, but I think that it's some skill you acquire with time and experience. Also, there is a wonderful paper from Leslie Lamport: How to write a proof. This paper is not so much about how to write an elegant proof, but rather how to structure your proof. However, once you have a nice structure, I believe it's much easier to make it more elegant. Also, I try now as much as possible to encode my proofs in a theorem-prover (such as Isabelle/Isar), as it helps me understanding for instance what proof steps I can extract and generalize as lemmas.