But surely these sort of meaningless examples with randomized numbers like at the site aren't that useful? I mean, a "real-world" example of matrix multiplication would be having a proper translation or rotation matrix and then visualizing how it maps a point to its image.
That demonstrates multiplication of a matrix with a vector which is being rotated, but multiplication of two matrices corresponds to composition of the corresponding operators.
This is actually pretty instructive. For instance, rotations in three-dimensional space do not commute in general; hence neither does matrix multiplication.
Technically matrix-vector multiplication is just a special case of matrix-matrix-multiplication, but I take your point. Indeed it would be illuminating to see how exactly matrix multiplication composes two transformations.
