How does one go about earning the Proofreader badge? I have attempted looking at other answers and couldn't find much.
Is there a detailed process that could be explained please?
Approve or reject 100 suggested edits.
This means that -once you gain enough rep - you will have to work on this review queue (available from the link on the top left of the site header and perform at least 100 successful reviews.
The access to the Edit Suggestion review queue is tied to other privileges you can gain with enough reputation points. In this specific case, you should at least have the Edit Post privilege - which is available once you reach 2000 rep point on a site (notice that to review suggested edits to tag wiki pages you need the Approve tag wiki edits privilege, available at 5000 rep point. So until that, you will have access only to a subset of the available reviews).
Final disclaimer: all the links in this answer are relative to this site, so please refer to the specific pages on your site of interests if you need more details or want to check your progress.
A little clarification since it was asked in comments. First, yep, it is technically possible to get the badge before reaching 2000 rep score - but that means all your reviews must come from suggested edits to your own posts. Unless one tries to game the system, that probably won't happen very often.
Second... the Badge page lists all the available badges including ones that you can't get yet (example: Citized Patrol if you don't yet have the ability to flag posts) or ones that can't be awarded anymore (example: Precognitive for a site that is already past its commitment phase).
It is worth noticing that the badge tracker tool available in the user profile page does instead provide the ability to filter out not-yet obtainable badges.
To access the badge progress tracking selection window click on the small gear icon shown on your user profile near the badge name.
As you can see, you have multiple tabs. The last one, "Available" only list badges that you are actually eligible for at the moment.
In that merit it is worth noticing that the Proofreader badge is indeed shown on the available tab for <2k rep users because as we said before it is still technically available to them, just very hard to get without gaming the system.
You'll first need to earn the privilege to review Suggested Edits, which is at 2000 reputation (or 1000 reputation for Beta sites). Then, you'll gain access to the Suggested Edits review queue, where you have to complete 100 review tasks.
Since you're always allowed to review suggested edits on your own posts, regardless of your reputation, it is in theory possible to earn the badge without 2000 reputation. But most of the times, those 100 posts (or slightly less if some posts are edited multiple times) will already earn you the reputation if they're at least moderately useful. This SEDE query shows one user who is almost halfway (43/100) this scenario, though.