The cosmic microwave background provides a convenient reference frame for measuring motion (called the co-moving frame). If you are moving relative to the CMB then the doppler shift means the CMB looks slightly hotter in the direction you are moving (the dipole anisotropy), and slightly cooler in the other direction. This motion is called the peculiar motion. If we measure the CMB from Earth we find that in fact we are moving relative to the CMB, but at only 368 km/s (you might think 368 km/s is pretty fast, but it's peanuts compared to galactic red shifts) so we conclude the Earth is roughly stationary wrt the CMB.
General Relativity, or more precisely the solution called the FLRW metric, predicts that every star/galaxy/whatever finds itself to be roughly stationary wrt the CMB, so the fact that the Earth appears to be stationary is no surprise.
But if the red shift is caused by galactic motions then those galaxies would not be stationary wrt the CMB. You'd have to conclude the Earth is stationary and all the galaxies are moving away from it i.e. have non-zero peculiar motions, and you'd have to conclude that this peculiar motion increases with distance from the Earth. This would make the Earth a very special place, which seems improbably because the Sun seems to be a fairly standard star, in a fairly standard galaxy, in a fairly standard cluster, and so on. It's much more likely that the red shift is due to the expansion of the universe and not peculiar motions of galaxies.
So given that it's extremely unlikely that the Earth just happens, by chance, to be at the centre of the universe, and that we have a well tested theory (General Relativity) that predicts the expansion of space, few of us doubt that the expansion of space is the cause of the red shift.