The physics is actually much easier than it seems at first glance.
Power generators are engines just like the everyday ones we see all around in our cars, lawnmowers, snowblowers, etc.
Except for new power sources like some wind and solar systems with electronic inverters, the vast majority of power is supplied by large rotating AC generators turning in synch with the frequency of the grid. The frequency of all these generators will be identical and is tied directly to the RPM of the generators themselves, generally 3600 RPM for gas turbines and 1800 RPM for nuclear plants. If there is sufficient power in the generators then the frequency can be maintained at the desired rate (i.e. 50Hz or 60Hz depending on the locale).
The power from the individual generators will lead the grid in phase slightly by an amount roughly corresponding to the power they deliver to the grid.
An increase in the power load is accompanied by a concurrent increase in the power supplied to the generators, generally by the governors automatically opening a steam or gas inlet valve to supply more power to the turbine. However, if there is not sufficient power, even for a brief period of time, then generator RPM and the frequency drops.
This is much like what happens to a car on cruise control if you start going up a hill, if the hill is not too steep you can maintain speed, once you reach the limits of the torque supplied by the engine, the car and engine slow down. If the combined output of all the generators cannot supply enough power then the frequency will drop for the entire grid. All the generators slow down just like your car engine on a hill.
For large grids the presence of many generators and a large distributed load makes frequency management easier because any given load is a much smaller percentage of the combined capacity. For smaller grids, there will be a much larger fluctuation in capacity as delays in matching power supplied are harder to manage when the loads represent a relatively larger percentage of the generated power.
So a battery systems like the one in the article is really designed to keep short-term fluctuations in power requirements from dropping the frequency because of lags in the governors and generators which require a finite time to adjust to the new power requirements. These "frequency regulator" power stations can supply very high power for short bursts to keep the power requirements even so that the other generators don't see too much load faster than they can respond due to mechanical limitations.