Why does stator current increase with increasing load in a synchronous motor?
Since the motor converts electrical energy to mechanical energy, the electrical input power must be equal to the mechanical power transmitted to the load plus power lost in the motor. Since the input voltage is constant, that would have to be reflected in the input current and power factor since power = voltage X current x power factor.

The mechanism by which electrical power is converted to mechanical power is explained using the equivalent circuit of the motor. Just as in a DC motor, a back EMF is generated in an AC synchronous motor. So the motor equivalent circuit is an AC back EMF generator in series with the internal impedance of the motor. The back EMF opposes the source voltage so that the stator current is proportional to the source voltage minus the back EMF divided by the internal impedance of the machine. With an AC machine, the voltage, current and impedance values are all complex numbers. The phase angle difference between the terminal voltage and the back EMF is determined by torque angle, the angle between the rotating stator and rotor magnetic fields. As the name implies, the torque angle is proportional to torque.


  This leads me to another question which might help: Does the rotor's magnetic field of an induction motor affect the stator's current value?


I don't think that is related to the synchronous motor question. The induction motor rotor current is ultimately supplied by the stator, but the magnetic fields in both the stator and rotor are pretty much constant as long as the applied voltage and frequency are constant and the ratio of voltage to frequency is constant.