How can I best edit a paper to help get it published?
You've already talked about reading up journal articles in the field, so I'll skip that. On top of that, there are a few ways.

Follow the journal's format guide

Ask your friend which journal is the next target. Go to visit the journal website and look for the "instruction to authors." You can find format-related instruction there. A format compliant article is less likely to trigger an instant rejection/return.

Read about scientific writing

There are a few guides that I consider pretty useful:


The Craft of Scientific Writing by Alley is perhaps a classic for engineer-type of writing. It also provide a good collection of tips and gems for different sections.
Essentials of Writing Biomedical Research Paper by Zeiger is a wonderful desktop reference for biochemical type of writing. It also provides a lot of good vs. bad examples.
Writing Science: How to Write Papers That Get Cited and Proposals That Get Funded by Schimel is a bit of a black sheep. It does not teach you how to write, but it gives an excellent account on how to chain up or arrange ideas for maximal impact, done in the levels of the whole paper, to section, paragraph, sentence, and syntax. It also draws heavily from techniques used in fiction writing, which is quite intuitive.
The Craft of Research by Booth et. al. does not purely focus on writing, but also discusses how to set up arguments, present concepts. It may be a bit more hands on for you, probably more suitable for your friend who is doing the writing.
A Manual for Writers of Research Papers, Theses, and Dissertations, Eighth Edition: Chicago Style for Students and Researchers (Chicago Guides to Writing, Editing, and Publishing) by Turabian is an overall very useful desktop reference. It complements the Craft of Research.


Talk to the specialists

If the paper is really that controversial, I think you should talk to some people who have a good command in that particular field and get a gist of how to present or package the ideas with maximal chance of being considered.

Hire a professional editor

It's also prudent to know your limit. If you feel this is too much, then you should ask your friend to get help from the institute's English language support or hire a professional scientific editor. Editors can come with different specialties, some are experienced in medical writing, some are in science. Check their portfolio and try to match the article type as best as you can.

I said this because there is a problem in your question, if you feel that you're not capable of editing a scientific paper, how come you feel confident to evaluate his work with certainly such as: "My friend wrote a fantastic paper in their scientific field. I believe it is truly ground-breaking but it calls a lot of existing theory into question. If he's correct it will force many accepted articles to have to be rewritten?"

I don't mean to be insulting, just wish to point out that professional works sometimes are best left to professionals, especially when we don't have time to become one.

Best of luck, and I wish your friend a successful publishing process.



Disclaimer for everyone:

When reading/evaluating my answer, please be mindful that in no way I am agreeing that the paper is ground-breaking or fantastic. I merely provide resources to the questioner to improve his/her ability to comprehend and edit a scientific paper.

Whether someone with limited experience or capability can do ground-breaking work is not in the scope of this answer, and I have no comment either way. I just want to point out that I have not read the paper so I can't comment.