First, a disclaimer: I have done some work in this area (I've collaborated with some of the authors of the second-generation proof), but I'm not an expert, and frankly I don't actually know the answers to some of these questions. Take everything below with a pinch of salt.
> I don't get why it's simple. These papers did not explain it.
This is somewhat obscure: if you look at Griess' *The Friendly Giant*, there's a proof that the group is simple in section 12. The basic idea is that reducing modulo p leaves a simple group of the same order for any prime p (other than 2 or 3), which implies that they're all exactly the group that you started with (there's some heavy-duty classification machinery involved in both steps of that, and I don't know that there's a nice exposition of it in existence). These papers do enough work to show that their group is isomorphic to Griess', so they don't have to repeat all of that work.
> Its construction makes no intuitive sense, it just looks like a bunch of calculations that randomly work.
Yeah, pretty much. The objective is to make something that's simple, not intuitive. The intuition building was sort of backwards: there was a good reason to think that there would be a simple group of this order, and some known properties about it, so people just tried to construct a group satisfying all of those properties, which turned out to work.
> Everything starting from the Parker loop looks like mystery.
> Everything starting from the Parker loop looks like mystery. I don't understand Moufang loop and I'm not sure what happen if I replace the Parker loop with another Moufang loop.
You end up with the wrong order, or you end up with something with an "obvious" normal subgroup.
> Basically, I'm missing the big picture.
I'm not sure you are. These constructions are designed to have the properties that they need to have, and that's kind of it.
> I don't know why it's sporadic. It depends on Golay code, Leech lattice, Parker loop, which are exceptional objects, but I'm not sure where these exceptional properties are relevant.
It's really unsatisfying, but there isn't much hope for a more satisfying answer (being sporadic is a negative property, so there's not much hope for a good reason why things are sporadic): its order doesn't let it sit in any of the infinite families.
> I don't understand its relation to string theory. Apparently I need to learn Vertex Operator Algebra, which is a deep subject on its own.
I also don't understand this, but I'm led to believe it's via monstrous moonshine, somehow.