The key thing here is that the generating plants are not indepedent of each other. They are all connected to the same national (or even international) electricity grid.
Sure, if the demand drops drops far enough, you shut down some of the generating plants completely. That's why you have a mixture of large plants (e.g. nuclear or large coal-burning steam generators) which are efficient but can only respond slowly to changes in demand, and also smaller generators (e.g. natural gas fired) which can be stopped and started lilterally in a few minutes.
In simple terms, you can think of this like the cruise control on a car. You set the speed you want to travel, and the system automaticalliy adjust the throttle to burn more or less gasoline, or even apply the brakes, depending on whether you are on level road or going up or down hill.
If one electricity generator on the network tries to supply "to much" power, because all the generators are connected together what happens is that ALL the generators will tend to speed up a little bit, not just the one that is working "too hard". The control system works by keeping the frequency of the supply fixed at its correct value (50 or 60 Hz depending what country you are in). That frequency corresponds to the speed in a cruise cnotrol system.