Publishing a review of another person's proof For a Graduate level math course one of our assignments was to prepare a lecture on a topic to give to the rest of the class.

My topic was to present a result that had been discovered some years ago as it would compliment some of the material from class. I wrote up a small paper on the topic to help prepare me for the lecture.

I think it would be useful to publish because there are only two places I know of where the proof can be found: the original proof is in a Swedish mathematics journal, and the other is in a physics textbook and lacks a certain amount of rigour or intuition.

My lecture followed the textbook's presentation and I added to the proof some rigour, and some intuition behind the steps of the proof. However, I didn't really add anything to the result, just explanations and flushing out omitted details. 

Would it be considered plagiarism if I were to publish a review of their proof on a website like Arxiv? I don't think the review is important enough to be put in a real journal, but I do think that some people might find the explainations useful.

Another thing to note is that this would be my first "publication", could it hurt my future academic prospects? Thanks for the advice!
If you overtly title your piece something like "explication of X's proof of Y", and give the citation, (and do not literally copy verbatim swaths of their article) this is a completely fine thing to do. Or give it a more explanatory title, and in the text immediately say that you are simply expanding or elaborating on the proof in the original source(s).

It will absolutely not hurt your future, because the people who are "strict" about what "publication" supposedly means will ignore anything that appears only on arXiv, because, in their minds, it was not "published" at all, anyway. And, to many people, taking the trouble to digest an important thing and re-tell it for the benefit of others is a wholesome activity.