Hello Semusal,
I will make it simple for you, imagine you are driving a car and you want to maintain its speed. Then you face an incline (which is an extra load), if you kept providing the same amount of fuel to your car's engine, the speed of the car will reduce at the incline, therefore you will need to increase the fuel inlet for your car to maintain the speed that you desire. The generator works in the same exact concept, if load increases on the generator, with the same fuel inlet to the generator's turbine will not be enough to maintain its speed and therefore it will slow down and lower the frequency. Therefore, the governor will react to the increase of the load and push more fuel into the turbine and increase its speed to maintain the desired speed and frequency. IF the generator is unable to maintain the frequency as the load increases, then the generator's capacity is not enough to supply the total load and it will trip on under frequency (load shedding will be required in this case to avoid tripping of generator).

The same idea can be applied for voltage, if the load added is an inductive load, it will affect voltage of the generator (lower it). Because the generator has to supply that reactive load and the AVR will have to increase the field excitation to increase the voltage and maintain it.

In Summary, in island operation (isochronous mode) the generator has no control over the load's real and reactive power and therefore has no control over power factor. The generator can only control frequency and voltage to maintain them according to the power and reactive power requirement from the load. But be careful, this is the case with island mode and it is not the case when connected in parallel with other generators or to a big grid.

Also note, that AVR will control voltage or reactive load (or power factor) depending on the mode of operation. Governor will control real power or speed (frequency) depending on mode of operation. I hope I answered your question.