The underlying assumption is that the generator is connected to a (large) network. This means that the voltage at the terminals of the generator is fixed, that is, both magnitude and phase are fixed. Let's call this voltage U. The phase angle can be selected to be zero.
In addition, the power of the generator is fixed, determined by the machine turning the generator (the "prime mover"). Only the magnetization current can be controlled. It determines the magnitude of the voltage induced in the stator windings. The phase of the induced voltage cannot be independently controlled. The induced voltage is E cos(fii) + jE sin(fii), where E is the magnitude of the induced voltage and fii is the phase angle.
The current in the stator winding is I = (E cos(fii) + jE sin(fii) -U) / jX , where X is the generator reactance. It is assumed that the resistance is so small that it can be neglected, and that the windings are connected in a wye. The complex power of the generator is equal to the product of the voltage times the complex conjugate of the current, multiplied by three, because of the three phases, S = 3UI*. The real power is the real part of the complex power, P = Re(S) = 3 U E sin(fii)/X.
When the magnetization current is increased, the magnitude of the induced voltage E increases. But because the power is fixed, the phase angle fii must change so that the product E sin(fii) stays constant. Because the induced voltage changes, the current (phase and magnitude) will also change. The reactive power and the power factor can thus be adjusted in this way by changing the magnetization current.
The interesting part is that sin(180deg - fii) = sin(fii), so that the same power can be obtained with two different phase angles, in principle, at least. I do not know, how this is achieved in practice. Maybe someone can explain this?
