I'm not sure if this is useful but for what it's worth, this is my experience. I did a totally free 10-day silent meditation retreat (Vipassana Meditation) a few years ago -- the only way to learn this particular method is to do the course.

I did that a few years after dealing with severe depression (for which I'd been on meds). I know that I couldn't have done it when I was in the midst of that depression -- also, in the course application form, they specifically ask about mental health (including episodes, meds etc) and I'm guessing they filter based on it. That sort of makes sense since it was just incredibly intense -- a few people broke down and one of my room-mates who was a trained psychologist was pissed as heck that they were doing such intense stuff without much more screening for mental health or more importantly, having a psychologist or mental health professional on call.

So, it's a catch-22 off sorts -- meditating can help with mental health but there needs to be some baseline level of mental health I guess for it to work or at the very least, not make things noticeably worse. I've found this to be true even outside of the course boundaries (and the people running the course also advocate this) -- if one is too agitated or dealing with some seriously profound stuff (like severe depression or anxiety in my case), meditating is really really really not recommended. For instance, even now, when I'm in the throes of some intense bout of anxiety, I take a walk or do something else but the last thing I'll do is meditate.

This is because meditation, at least for me, makes me aware of all kinds of emotions that I'd otherwise successfully suppressed (or were very well compartmentalized as my therapist would say). So, sure, while meditation is supposed to give me the ability to observe this stuff, sometimes I just can't muster up enough of that strength to observe and instead I'm finding myself facing all this painful stuff in a way that feels overwhelming.

Having said all that though, doing that course and learning that technique has made all the difference to me. I don't consistently meditate for the stipulated 2 hours a day (and in fact, I sometimes deliberately take a break -- more on that below) but just having done that often enough, I know that things have shifted for me and I am aware of a process by which I can make profound mental / emotional shifts. That by itself is super helpful.

To answer your two specific questions --
1. The way in which I found that it makes a change is by just making me feel like I have a lot more space within me. That feeling of space opens up options, choices etc that I otherwise wouldn't notice. So, I don't as much feel that ragged, jagged, on-edge, constantly-intense feeling that I have otherwise (combination of anxiety, depression etc etc). But, again, the caveat, I can only do this or practice, when I have a baseline level of okay-ness.

2. Why not distract yourself? Sure, I do this often and I won't say that it doesn't work. The way though that meditation feels different to me is that it changes stuff for me at a deeper level than the distraction -- so, for instance, I could be relieved a bit of my anxiety about something by say surfing the web and at the end of it, I could even feel like I have a bit more space and can think more calmly. But, that feeling (both in terms of depth as well as the length of time it works) is likely to fade and also isn't as profoundly helpful to my ability to manage my emotions. I don't know how else to explain it -- it's not something I think can actually be explained (as frustrating as it is to use the cliche, it really feels like one of those things that has to be experienced for oneself).

To reiterate though, I'm one of those people who won't advocate meditation as a catch-all cure -- there's a New York Times article on it as well which argued that meditation, at times, can make things worse.

I've also found that it tends to add to my ability observe rather than feel stuff. When I started therapy then and was meditating pretty consistently, I realized that I couldn't actually summon up my emotions easily in therapy.

This was because as my therapist pointed out, one of the ways I'd dealt with stuff (what she called extreme abuse) growing up was to observe myself -- she said that most people do it at times but it had become my default mode of functioning (rather than feeling). And, in some ways, I think meditation strengthens this part of me -- super helpful a lot of the time but not always.

So, I actually quit meditating for a pretty long time (months) in order to show up and be emotionally present (at least to the extent I could) in therapy -- it was seriously a gawdawfully painful exercise but I think it has helped a great deal.

Also, my therapist had early on commented that I tended to use meditation to regulate my emotions -- for some reason, it stung me (mostly because it was true). So, I think that was another reason that I quit meditating for a while -- I wanted to figure out what was going on inside of me without sort of checking out of it, so to speak.

So, to sum up, I don't think meditation straight up makes a difference to my depression or anxiety -- as in, I can't say I'm cured of it by meditating. And, in fact sometimes, it actually makes it worse.

But, it does help in a lot of ways -- both broad and deep -- that have been truly beneficial.

Reading my post now, I'm not at all sure if this will give you the answers you were looking for