The OP wrote:
I am trying to understand what he means by "why".
Although several respondents have expressed skepticism that this question can be answered satisfactorily, I maintain that it's not that hard to understand. For doubters, I would offer the following paraphrase of Conway's statement, "I would really like to know why the Monster Group exists":
I would like a simpler and more natural construction (or existence proof) of the Monster.
Justification for this paraphrase may be found in Conway's own work in finite group theory. For example, what was the point of his paper, A simple construction for the Fischer-Griess monster group? Conway himself wrote, "The main aim of this paper is to present a simplified construction [of the Monster]." If you read his book with Sloane, Sphere Packings, Lattices and Groups, you will similarly find an emphasis on the nicest possible constructions. In Chapter 11, we read:
The Golay code $\mathscr{C}_{24}$, the Steiner system $S(5,8,24)$, and the Mathieu group $M_{24}$ are beautiful combinatorial objects with a great wealth of structure and applications. The MOG [Miracle Octad Generator], and its companion-at-arms the hexacode, are computational tools that enable one to perform mental calculations on these objects with great ease. … There is also a MINIMOG, with companion the tetracode, whcih together perform similar services for $M_{12}$.
In the introduction, we get an example of something that the authors regard as shedding light on the miraculous properties of the Leech lattice:
Considerable light is thrown on these mysteries by the realization that the Leech lattice and the Niemeier lattices can all be obtained very easily from a single lattice, namely $\mathrm{II}_{25,1}$, the unique even unimodular lattice in Lorentzian space $\mathbf{R}^{25,1}$.
Again, ease of construction is regarded as very important. A similar dissatisfaction with ad hoc miraculous constructions and extraneous detail can be found in all of Conway's work, not just his work in group theory; a case in point is his proof of the classification of closed surfaces, which he dubbed the "ZIP proof" or "Zero Irrelevancy Proof."
The sources cited by the OP corroborate this point of view. The IAS article says, "In his view, conformal field theory is too complicated to understand, and thus too complicated to be the only answer." Again, note the emphasis on simplicity versus complexity.
So I think we can safely assume that what Conway was hoping for was that in the next 100 years, a much simpler construction of the Monster would be found.
Regarding the question of why the existence of the Monster couldn't be a "coincidence," I feel confident that Conway would have rebelled against that attitude. It is true that there are some types of mathematical statements that Conway might have conceded are true by accident. As I mentioned in that other MO answer, Conway felt there was some force to the heuristic, probabilistic argument that the Collatz conjecture is true, but not provably so. But in that setting, the culprit is that provable intractability (and lack of structure) is crouching at the door. In the case of the Monster, we have a proof of its existence, and there is tons of structure present. I am sure that Conway would have categorically rejected any argument along the lines of, "Oh well, you know, if you just generate group multiplication tables at random, it's not at all surprising that a simple group of order 808, 017, 424, 794, 512, 875, 886, 459, 904, 961, 710, 757, 005, 754, 368, 000, 000, 000 would emerge just by chance. Let's move on; nothing to see here."
I seem to remember having an exchange with Conway (maybe on the math-fun mailing list many years ago?) in which I raised the possibility that the main theorems of class field theory possess an ineliminable core of miraculous coincidences that can't be "explained" in a purely "conceptual" way, and Conway said that he thought that attitude was too defeatist, and that we shouldn't stop trying to find increasingly simpler proofs of something so beautiful. Certainly, Conway's philosophy of doggedly seeking simpler explanations of mathematically significant facts served him immensely well over the course of his career.
EDIT (October 2023): I just discovered a talk by Richard Borcherds from a couple of years ago. Someone asked a question about the construction of the Monster, and Borcherds' answer may give some insight into what Conway was hoping for.
This is an open research question for anyone who's interested: Find a natural construction of the Monster. Every construction of the Monster we have is a mess. Most of the constructions are essentially simplifications of Griess's original construction, and the problem with all of them is, you take two huge vector spaces which seem to have nothing to do with each other; you take their direct sum, and you put some sort of piecemeal algebra structure on this, and by some amazing fluke it just happens that the Monster is a group of automorphisms. You would never find this construction unless you already suspected the Monster existed and were trying to construct it. What we'd really like is a sort of one-piece construction where you construct it just in one piece. For instance, [for] Conway's group, you take the Leech lattice in 24 dimensions, and that's a very natural single object in 24 dimensions. We don't have anything like that for the Monster yet.
