I finally read up on the construction of the Monster group. Now what?

Maybe I'm wrong, but I don't know if you're going to find anyone here that knows enough about finite group theory, and especially this part of it. It's famously esoteric and ungodly in the sheer size of the proofs or constructions.
I don't know anything about finite group theory, but just as a general comment, most research-level papers are aimed at experts in the relevant field and will not provide enough narrative scaffolding to be truly understandable to outsiders. As in, yes, you can technically follow the proofs line by line, but the main conceptual threads are going to be difficult to pick out.

I recommend instead trying to find a textbook treatment or an expository article. I'm not sure if one exists though.
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.
Those look like good questions to ask the author of that second paper. The topic seems specialized enough that people working in the area might really be the only people who can answer such questions. Even if you find the questions too basic to be worth asking, it doesn't hurt to ask, at the very least, but actually I think many academics would love an opportunity to talk about their work to outsiders, and your having read the paper demonstrates a lot of good will on your part and provides a shared context that would ease the discussion greatly.
As I understand it, the Monster group is tough to work with not because it's so large, but because all of its irreducible linear representations over ℂ are very high-dimensional (contrast the compact Lie group SU(2) which has cardinality the continuum but which has irreducible representations of dimension m + 1 for every nonnegative integer m). The representation theory of the Monster is thus very difficult to get a handle on.

One would hope to characterize the Monster as a group of automorphisms of some compact complex manifold M, along with some holomorphic line bundle L on M such that the induced action of the Monster on the sheaf cohomology groups of L realizes all of the Monster's irreducible complex representations. (This is analogous to the philosophy behind the Borel-Weil-Bott theorem). This would give a very geometric way of constructing the Monster and its representations.

This hope has not been realized, and if such a complex manifold exists, its construction is very difficult. Because we have no nice interpretation of the Monster as a symmetry group of some natural geometric or combinatorial object, all methods of constructing it are going to seem complicated and ad hoc.
“it just looks like a bunch of calculations that randomly work”. Seems to me that you summed up research level math pretty well there.
For some motivation,  Richard Borcherds's YouTube channel is fantastic.  Specifically the videos on Sporadic groups and Monster Moonshine might interest you.
Frenkel meurman and lepowsky's "vertex operator algebras and the monster" provides some context.

They define vertex operators in a chain of generalizations from representations of affine lie algebras to representations of vertex operator algebras.

Then they go over lattices and define the leech lattice, explain its important properties and use it to define the moonshine module and the monster.

It won't answer why the group is sporadic, and you may feel that computing the dimensions of the pieces of the moonshine module and comparing with the j function is just replacing one coincidence with infinitely many coincidences, but it's still something.
Aren't both Richard Borcherds and John Conway quoted as saying they don't understand these questions?

Then I think it's pretty fair to say that no one knows why the monster is the way it is.

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