Essentially it allows one to relate space and time averages.
One example for which it has been proven is for geodesic motion on compact hyperbolic spaces. Its failure for bodies interacting gravitationally is important, as if this property were present the Earth's orbit would be highly disturbed by the other bodies in the solar system. Birkhoff proved it holds in certain conditions. The hypothesis that statistical mechanical systems have this property was made by Boltzmann, and now it is a branch of mathematics that studies group actions on measure spaces. Essentially it allows one to relate space and time averages. FTP, what is this hypothesis that phase space trajectories eventually pass arbitrarily close to any given point?
One example for which it has been proven is for geodesic motion on compact hyperbolic spaces.
ergodicity