Gravitational red shift vs Doppler redshift: Is the universe really expanding?
Ultimately this is an application of Occam's razor

In order for the light to be gravitationally redshifted, it would have to be coming out of a deep gravitational well. For the red shift observed in galaxies to be gravitational, you would have to suppose several things. 

First, that the stars in distant galaxies are somehow much denser: more than neutron star dense. (or possibly that entire galaxies are as dense as neutron stars.) A neutron star has a redshift of about z=0.35 distant galaxies have redshift of more than 7. No known object has a gravitational redshift like that.

Secondly that the density is proportional to distance from us. Placing us at a special position in the universe. 

It is much simpler to interpret the redshift as a doppler redshift, and therefore an expanding universe