We need more people teaching math through visual intuition. Life a friend of mine said, "if you want to do computation fast, phrase it as a problem for your GPU, er, visual cortex".
Also, remember that back propagation is a very general algorithm, it works not only on linear transformation weights but on any direct acyclic computation graph that is differentiable in its weights.
Here is a tool you can play with to visualize this: http://playground.tensorflow.org
If you liked this video, try this different visual intuition of what a neural network does that I find even better:
http://colah.github.io/posts/2014-03-NN-Manifolds-Topology/
Also, remember that back propagation is a very general algorithm, it works not only on linear transformation weights but on any direct acyclic computation graph that is differentiable in its weights.