The excitation system requires a very small fraction of the total power being generated. If we could simply increase the excitation (a very small amount of power) and increase the generator's real power output, the world's energy problems would be solved, because we would have a perpetual motion machine.
In the case of a generator connected to a large grid, the generator will inject any desired amount of power into the grid if its prime-mover is fed the desired power (plus a small additional amount of power to take care of losses). This is true, regardless of the total load on the grid, because the generator's output is an extremely small fraction of the total grid power, and it alone cannot make drastic changes to the grid's frequency.
Normally, the load varies by a very small fraction of the total grid power. If the load increases, the frequency of the entire grid (including the generator in question) lowers a very small amount, generally less than one-hundredth of one Hz. The frequency slew (that is, the rate-of-change of frequency) is very low, because there is a massive amount of energy that is stored as the kinetic energy of the rotors of all of the generators. At this point, nothing needs to be done; the system simply runs a little faster or slower.
Over time, as the load changes a greater amount, the frequency moves further from the nominal frequency (50 Hz or 60 Hz). When the difference between the actual frequency and the nominal frequency becomes greater than about 0.01 Hz, action is taken to make changes to the output of the grid's generators.
The specific action may be determined by the regulating authority (for instance a power pool in the US) and it is usually based on economics, subject to other constraints. If the load has increased (and the frequency is less than the nominal frequency), the generators that have the lowest incremental cost of power will be asked to increase their output, or if all generators are near their limits, new generators (with the lowest incremental cost) are asked come on line. It's important to note that a generator's limit is usually 80% or 90% of its rating. The 10% or 20% of unused capacity is the system's "spinning reserve", which is used to maintain grid stability for sudden, large power variations.
The same thing happens with a generator connected only to its load or a weak grid with just a few other generators. However, because there is relatively little kinetic energy stored in the rotors of the one or few generators, the change in frequency associated with a load change is much greater, so frequency variations are much greater and corrective actions may not be implemented before the frequency varies by more than a few Hz.