To such a complex problem, there cannot be unique answer. I see many, which all justify the tremendous interest that mathematicians have devoted so far to matrices, rather than to hypermatrices.
Ubiquity. Matrices are used by every species of mathematicians, and beyond, by a large fraction of scientists. This is perhaps the only mathematical area to enjoy this versatility. Let me provide a few examples. Matrix exponential is fundamental in differential equation (more generally in dynamical systems) and Lie theory of groups. Symmetric matrices are used in quantum mechanics, statistics, optimisation and numerical analysis; they have deep relations with representation theory and combinatorics (see the solution of Horn's conjecture by Tao \& Knutson). Positive matrices are encountered in probability and numerical analysis (discrete maximum principle). Matrix groups are used in representation theory, in number theory (including modular forms), in dynamical systems (because of symmetries). When depending on parameters, matrices enter in PDE theory as symbols.
Simplicity. The concept of matrix is by definition simpler than that of hypermatrices. It is natural that the study of matrices precedes that of HM. This argument will fade as time increases, of course.
Richness. What makes a field particularly attractive is that it involves several apparent unrelated concepts in order to produce unexpected results. This happens in matrix theory, because on the one hand, we may view them as linear maps (where conjugation is relevant) and on the other hand we may see them as bilinear or sesquilinear maps (where congruence is relevant). It becomes especially fruitful when we go back and forth between both points of view. This happens in the remarkable theorem that normal matrices are unitary diagonalizable, but also in the parametrization of a Lie group by its Lie algebra via the exponential and the Hermitian square root. I am not at all aware of the theory of HM, but if they do not form naturally an algebra, I doubt that their theory could be so rich, or if it is, it will be for completely different mathematical reasons.
To temperate this pledge, let me say that hypermatrices have been studied (although not so deeply) under the name tensors. They are of great importance in differential geometry (Ricci curvature tensor, with the many identities named after Christoffel, Gauss, Codazzi, ...) and in its applications: general relativity, elasticity. These are undoubtedly difficult topics, where even simple problems are not well understood. To mention one of them, there is still no satisfactory description of the twice-symmetric tensors of fourth order ($a_{ijkl}=a_{jikl}=a_{ijlk}$) that satisfy the Legendre-Hadamard condition $$\sum_{i,j,k,l}a_{ijkl}x_ix_j\xi_k\xi_l\ge0,\qquad\forall x\in\mathbb R^n,\xi\in\mathbb R^d.$$ It seems to me that the use of HM is too scattered, and therefore there is no research community specializing on all their aspects. Edit. Likewise, the notion of rank, although correctly defined in the case of tensors, is hard to manipulate and to compute explicitly. This is the reason why the exact algorithmic complexity of the multiplication of matrices is still not known (the operation $(A,B)\mapsto AB$ in $M_n(k)$ may be viewed as a $3$-tensor, and its tensorial rank governs the number of operations needed in an $n\times n$ mulitplication).
