1 The mental image I have, which may or may not be helpful to anyone else, is of a row of the LH matrix sliding over and rotating clockwise until it matches a column of the RH matrix.

2 If you don't mind working it out from scratch and have a more "algebraic" than "geometric" brain, the following considerations may help:

• Matrices are indexed row first, then column (think "reading order": if you think of the indices as numbers, then as you read along rows like reading a book they increase monotonically).
• When you multiply matrices, what happens is that "middle indices match": element (i,k) of the product is sum over j of left[i,j] right[j,k]. See how the j's match up and "cancel out".

3 The product matrix is an array of inner products of rows from the LH factor and columns from the RH factor.

I make no guarantee that any of those will help, but if any of them seems to be heading in the right direction but not getting there I'm happy to try to adapt further...