These quotes are from pages 89 and 90 of Velleman's "How To Prove It". If only I'd read all this when I was first introduced to a proof, I wouldn't have been so stressed!
"When mathematicians quote proofs, they usually just write the steps needed to justify their conclusions with no explanation of how they thought of them."
"Although this lack of explanation sometimes makes proofs hard to read, it serves the purpose of keeping two distinct objectives separate: explaining your thought processes and justifying your conclusions."
"The primary purpose of a proof is to justify the claim that the conclusion follows from the hypotheses, and no explanation of your thought processes can substitute for adequate justification of this claim. Keeping any discussion of thought processes to a minimum in a proof helps to keep this distinction clear."
"Don't worry if you don't immediately understand the strategy behind the proof you are reading".
I could hug this book right now.