One application I really like is in machine learning: the eigenface algorithm.
After I understand the plain English concept, then give me the math notation and proof and applications.
The way it works today is often to simply leave out the first part and I believe this is why many people find it hard to develop intuition and a real understanding of the concept.
For man it's just there, you memorize it, you apply it, you take the result and simply have faith that it is what you need because some book/ prof said to solve problem X use eigenvalues.
Since scaling is a really easy operation to understand, the space generated by these vectors will be really easy to understand. Note that different vectors can be scaled by different amounts.
Now it often turns out that the space generated like this is actually the whole of the space under consideration and this really simplifies the linear map we started with. Hope that helps.