For understanding the mechanics of matrix multiplication, I found it useful to think in an analogy consisting of a grid of two layers of pipes; One -- the input-pipes -- coming in one direction, and the other -- the output pipes -- laid in an orthogonal direction. Then there would be "taps" in the cross-sections between the input and output pipes, representing the numbers (multiplication factors, really) in the matrix. I illustrated this in this little drawing: http://imgur.com/gallery/gBs64

The point then is that the taps (again, representing the matrix values) determine how much of each item in the input vector, that should be mixed into each item in the output vector.

This analogy has the limitation that the taps are allowed to enhance the flow, not just limit it, like physical taps would. That is, outputting more than 100% of the input :P Also, while this way of illustrating it may make some sense for matrix * vector multiplication, matrix * matrix would probably become a prohibitively cluttered image.