The Pauli exclusion principle states that no two electrons with the same spin can occupy the same energy state in the same volume. Once the lowest energy level is filled, the other electrons are forced into higher and higher energy states resulting in them travelling at progressively faster speeds. These fast moving electrons create a pressure (electron degeneracy pressure) which is capable of supporting a star!
In particular, electron degeneracy pressure is what supports white dwarfs against gravitational collapse, and the Chandrasekhar limit (the maximum mass a white dwarf can attain) arises naturally due to the physics of electron degeneracy. As the mass of a white dwarf approaches the Chandrasekhar limit, gravity attempts to squeeze the star into a smaller volume, forcing electrons to occupy higher energy states and attain faster velocities. At the Chandrasekhar limit, the pressure exerted by the electrons travelling at close to the speed of light becomes insufficient to support the star, and the white dwarf collapses into a much denser state.
Electron degeneracy occurs at densities of about 106 kg/m3.