Why does having magnetic material in the middle of a stator help generate a current?
The magnetic material doesn't increase the conductivity of the stator's coil.

The principle of any electrical machine is to use a rotating magnetic field to generate electrical coupling which allow for energy conversion. The stator's current is just a reaction to try counte-attack the induction in the stator's coil(lenz's law). 

The reason why these materials are call "magnetic material" is that either they can generate a very large B field for relative small current or that their electron's spin are naturally aligned creating a noticeable B field.