This is a cool example of what Bret Victor calls an "Explorable Explanation" [0]. That said, I feel that it's more important to understand how and why matrix multiplication corresponds to a composition of linear transformations than learning the actual mechanics of doing the computation. You can get good at matrix multiplication without knowing what is going on. I view that as a less valuable activity than learning about linear transformations (vector-space structure preserving mappings) between finite dimensional vector spaces.
[0] http://worrydream.com/ExplorableExplanations/