I first encountered eigenmagic in machine learning --- we were interested in the eigenvectors of the adjacency matrix of a graph. When the space that the matrix lives in is so abstract, 'real world' examples don't make it any easier to visualize what's happening.

If you really want to understand them intuitively, I reccomend plotting a few hundred matrices and their respective eigen- values and vectors. When you feel like you can predict what the results will look like at each frequency, then the stories about rubber bands and shiny coins will make sense.

Or maybe you're faster, and the stories helped you make the leap --- in which case, ignore me!