Imagine you're looking at a ball rolling around on a complicated curved surface defined by differential (non-linear) equations, which can be a tricky system to analyze.

If the ball is near some sort of a saddle equilibrium point, the theorem says that you can simplify things by flattening out a small patch of the surface if you are (very) near that point.

This is a lot easier to analyze using simple linear algebra tools, and still gives good results for predicting what happens next.