I believe it splits the signal in half, giving the signal a + and - signal. The 12AX7 family of tubes is really 2 tubes in 1. Alot of times tube layouts will say V1a or V1b for this reason. Think of a 12ax7 as 2 seperate tubes in one packaged tube which we call a 12ax7. one of those "tubes" gets the positive and the other gets a negative. This signal is then fed to the power tubes and allows them to operate in "push-pull".
If you look at the signal on an o-scope you see the signal going above and below the base line. The phase inverter allows the signal to do this. Without it the signal would either be completely below or above the line.
I could be wrong on this, but I believe that is the quick explanation. It has been awhile though.