How do journals gain and lose reputation?
TL;DR Top journals tend to have that reputation because they're regularly publishing top papers. Similarly, journals that have a reputation for being "merely" pretty good tend to mostly publish papers that are "merely" pretty good.

I think that far and away the most important reason that certain journals acquire reputations for being strong journals is that they publish very interesting papers that prove important results. For example, if I'm asked to justify my belief that the Annals, JAMS and Inventiones are among the very strongest of all journals in pure mathematics, I'm definitely not going to say anything about mathscinet quotients or impact factors. Moreover, even though having a strong editorial board is incredibly important, I wouldn't mention that either. Instead I would use the fact that off the top of my head I can list a large number of seminal papers that were published in each of these journals. Similarly, if all the papers that I've read in a given journal seemed boring to me (or perhaps not even 100% correct), then I'm not going to think much of the journal and probably won't submit any papers there, regardless of what the many other journal ranking metrics say about the journal.

Since you asked for examples of new-ish journals that have acquired very strong reputations without being the flagship journal of an important mathematical institution, here are two: Geometry &amp; Topology and Algebra &amp; Number Theory. These journals were founded in 1997 and 2007 respectively and have already acquired reputations for being amongst the very strongest journals in their fields. The reason for this, I am sure, is that they both regularly publish extremely interesting papers. How do they attract such strong papers? I'm sure that part of the reason lies with their incredibly strong editorial boards. Having an editorial board full of very strong, well-known mathematicians is likely to result in a steady stream of strong papers since one of the most important factors to consider when choosing a journal to submit a paper to is the editor that will handle the paper. If I have a paper that I think is very strong then I'm not going to want to send it to an editor that I've never heard of and that might not even know what sort of people will make good referees. Instead I'd try to find a journal (at more or less the appropriate level) where there's an editor that I know works in my area and whom I suspect knows plenty of people that will do a good job of refereeing my paper. Thus, if your journal has a board full of well-known people that work in trendy areas then it's not too much of a stretch to imagine that before too long you'll be getting papers submitted that concern trendy areas, at least some of which are very strong and are written by other well-known people in the area (which in turn means that people citing their papers are likely to consider submitting their papers to your journal).