Yeah, as the load gets beyond the generator, the frequency drops. And as noted the govener tries to maintain RPM to maintain frequency. When you have many generators interconnected, all their governers work to keep the frequency constant, and if it drops then they all try to speed up.
in a large interconnected system, if the frequency goes too low, or the rate of change of frequency is too great (depending on relay capability and setting), then relays will operate to shed load, to attempt to stop the decline in frequency.
If frequency dropping continues then an underspeed cutout will operate, and take the generator off-line.
In a large interconnected system, if that happens a lot you get a cascading blackout.
The opposite effect can also happen; if a large load goes offline then the frequency increases, and the generators need to back off, reducing the prime mover input.
We had a good excursion like that a while ago here in New Zealand, when the earthquake hit, pretty much all of Christchurch dropped off line, causing a large over-frequency excursion. The 'quake caused the transformers to shake, causing the oil to slash about inside. The large transformers have oil sensors, and they thought the transformers were out of oil, and commanded the transformer be shut down to prevent damage. Couple of seconds later the whole city is powerless.
Then the destruction to the underground cables began...