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.

Generally it's a pretty basic concept in analyzing any system, 1st step is to use a constant, if it doesn't fit then a linear model, etc. Can use different approximations depending on the circumstance, e.g. you can approximate sin(x) with x around 0...

Is there something special to it in this particular case?

Hartman-Grobman isn't so much about approximation error by choosing different simplifying models. It's about being able to do some analysis on the linear approximation with the results carrying over to the original non-linear system. In particular, it lets you categorize stationary points of a non-linear dynamical system, using only the linear approximation at the fixed points, which is awesome since analyzing linear systems is super easy.

My take on this theorem is that it is most interesting to know when the topological equivalence doesn't hold, i.e. linear approx of equilibrium point can be poor when there is the possibility of a limit cycle oscillation.