If you didn't get an answer elsewhere yet: You click the button and it sits pending until approved (whether its Ernest approving manually or an automated batchjob I don't know). Seems to be once a day or so, so could be either but that's just anecdotal. That's it tho. Click the button and wait, no other steps lol.
Either way, once approved the magazine will just appear in your moderated list on your profile (no notification or anything from what I've seen) and from there it's yours to do with as you want. I've claimed a dozen or so over the last day or so to delete (unused ones people sat on).