[this is an attempt to combine two previously given answers, mdup's video demo and my "path-sum" story, so it might help to refer to those.]
after watching mdup's video demo i started wondering how it relates to the "path-sum" interpretation of matrix multiplication. the key seems to be that mdup's hand-drawn picture of the matrix product AB wants to be folded up to form the visible faces of an oblong box whose three dimensions correspond precisely to the points i, j, and k in a three-point path (i,j,k). this is illustrated by the pairs of pictures below, each pair showing the oblong box first in its folded-up 3-dimensional form and then in its flattened-out 2-dimensional form. in each case the box is held up to a mirror to portray the effect of transposition of matrixes.
in the first pair of pictures, the i, j, and k axises are marked, and in the folded-up 3-dimensional form you can see how transposition reverses the order of the axises from i,j,k to k,j,i. in the flattened-out 2-dimensional form you can see how it wants to be folded up because the edges marked j are all the same length (and also, because it was folded up like that when i bought the soap).
the second pair of pictures indicate how an entry of the product matrix is calculated. in the flattened-out 2-dimensional form, a row of the first matrix is paired with a column of the second matrix, whereas in the folded-up 3-dimensional form, that "row" and that "column" actually lie parallel to each other because of the 3d arrangement.
in other words, each 3-point path (i,j,k) corresponds to a location inside the box, and at that location you write down (using a 3-dimensional printer or else just writing on the air) the product of the transition-quantities for the two transition-steps in the path, A_[i,j] for the transition-step from i to j and B_[j,k] for the transition-step from j to k. this results in a 3-dimensional matrix of numbers written on the air inside the box, but since the desired matrix product AB is only a 2-dimensional matrix, the 3-dimensional matrix is squashed down to 2-dimensional by summing over the j dimension. this is the path-sum- in order for two paths to contribute to the same path-sum they're required to be in direct competition with each other, beginning at the same origin i and ending at the same destination k, so the only index that we sum over is the intermediate index j.
the 3-dimensional folded-up form and the 2-dimensional flattened-out form have each their own advantages and disadvantages. the 3-dimensional folded-up form brings out the path-sums and the 3-dimensional nature of matrix multiplication, while the 2-dimensional flattened-out form is better-adapted to writing the calculation down on 2-dimensional paper (which remains easier than writing on 3-dimensional air even still today).
anyway, i'll get off my soapbox for now ...