Here is one description on it.

For intuition about polarizations, it may be helpful to think like this: the category of abelian varieties over a field k (say up to isogeny) acts like a full subcategory of the category of representations of a group G on Q-vector spaces. (To show the shift in thinking we can write V(A) when we think of A as like a vector space.) The category of abelian varieties is too small to admit anything like the tensor products that exist in the larger category of G-representations, but one can see some multilinear structures. In particular, the dual abelian variety is like a dual representation---or at least a dual representation twisted by a character. A divisorial correspondence between abelian varieties A and B (a sufficiently rigidified line bundle on A×kB---a biextension of (A,B) by Gm) is like a G-invariant bilinear form on V(A)×V(B). A polarization on A is a divisorial correspondence on A×kA satisfying a symmetry and positivity condition (ampleness of pullback along the diagonal), which is like a G-invariant bilinear form on V(A)×V(A) satisfying a symmetry and positivity condition. There are some subtleties in the translation; for example, the symmetry on side of A corresponds to antisymmetry on the side of V(A). We know from the early pages of books on representation theory that G-invariant bilinear forms satisfying positivity conditions are useful---they give us complete reducibility, for example---and Weil used polarizations in a similar way in [1955d]. The fact that abelian varieties are polarizable corresponds to something like the fact that G is a compact group, but the aforementioned subtleties mean that statement is not quite right.

These vague remarks can be made precise when k=C in the way that Francesco Polizzi suggests: take V(A) to be H1(Aan,Q) equipped with its standard Hodge structure, and the above remarks correspond to some of the theory of theta functions (which is what Matsusaka had in mind in his introduction)