Before I begin, let me say I don't know anything about what I am asking. This morning for somewhat random reasons I decided to google moonshine and related things. As it were I discovered my ignorance and wondered why people study these things. I came up with a theory which I am sure is wrong, hence, why I am asking the question. My theory is that possibly in looking at SL(something) and thinking about CFT and what not, somebody noticed that some functions reminded them of exceptional objects may be some kind of isomorphisms between sporadic type things. They then said maybe if they hunted down exceptional isomorphisms they will say something about nice symmetries. Apparently there are some nice conjectures and ideas that have enabled them to bring "number" thinking into strings and (S)CFT. My ignorance of the topic is hopefully manifest in the format of my question. I am wondering if someone can give me a condensed version of the main purpose and idea of thinking about these ideas. As an aspiring physicist, it is important that I think about all ideas.
Actually this deep mathematics has already physical application. Indeed modular forms, Jacobi forms, the monster group (and the related moonshine theory) appear naturally in BPS black hole microstates counting.
The basic idea is that a black hole has a quantum structure. When gravity is strong we still don't know (at least not always) how to count and identify the microstates that give the entropy to the black hole. But in string theory black holes in the weak coupling limit can be seen as systems of intersecting branes. Thanks to supersymmetry this counting is protected from unknown corrections, therefore we can count the microstates from the brane picture and then extrapolate the result to strong coupling, where the black hole picture is correct.
Other applications are in the AdS/CFT correspondence. For instance, one can speculate on the identity of the gravity dual of the CFT with monster group symmetry.