This is a great question. Steve Brunton has some great videos about dynamical systems and their properties that are very accessible. This one I think does a good job showing the behavioral relationship between the eigenvalues and the underlying system: https://youtu.be/XXjoh8L1HkE

Recursive application of a system (model) over a "long" period of time gets rid of transients, so the system will fall onto the governing attractors of the system, which are generally dictated by the eigenvalues of the system. The recursive application also helps isolate the system so you are observing the model autonomously, rather than being driven by external inputs. This helps you tease out how expressive your model actually is versus how dependent it is on you feeding it from the target system's observations, which helps reduce over fitting and reduces bias.