Including proofs of known theorems in master's thesis
It would be a good idea to make it slightly more conspicuous that the theorem (and proof) are not original, e.g.:
The following theorem is due to [5]; for the clarity of our exposition we give a more detailed version of the succinct proof in [5].
This leaves no doubt in the reader's mind that the work is not original, and also explains why you chose to include the proof.
There are three reasons to include a proof in your master's thesis - two of them good, and one of them bad.
1: As part of your background section
If your work relies on important results in your field, including those theorems' proofs in your introduction and background sections makes sense. This is true even if the proofs are well known. A thesis needs to show that you understand your field thoroughly to your committee, and as a bonus, including your field's well-known results will make your thesis a good introduction to your topic for someone coming in from another field. Theses are actually read this way!
When you cite proofs in this way, there's no need to give an expanded proof. Paraphrase or quote the standard proof (citing it clearly) without much commentary. You're just giving an overview of what others have already accomplished.
2: Because the details of the proofs are important for your own work
In the sections describing your new contributions to the field, your work might depend on the specific details of a previous proof. Either the detail is directly relevant to your own proofs, or the intuition behind the proof you're citing is similar to your own approach. Calling out these specific details is helpful.
When you cite proofs in this way, it makes sense to expand them - but only by focusing on the specific details you want to discuss. Briefly describe the rest of the proof. And again, clearly cite the proof as it's not your proof, you're just commenting on it.
3. Because you want your thesis to be long and detailed
Part of good writing is knowing which details are relevant and concisely sticking to those details. Don't include well-known proofs just for the sake of padding out your thesis or because you're including proofs by default.
If you leave the proof in the main text, make it obvious that your work is not original (or at least that A proof of the thereom was done earlier). Prominent caveat.
You could also put it in an appendix.
I am less negative and more positive than Buffy on the benefit of showing this explication. Theses can serve a lot of purposes. Just make it clear that you are not claiming some discovery, but showing an exercise. But I think there can be benefit in it, both to you and to following lab mates--they will have the same issues dealing with the sparse literature that you did.