There's not much to see physically, but electromagnetically there's a lot going on. Once the rotor containing the field winding is up to near synchronous speed, the field is energized creating a variable strength electromagnet. This electromagnet is known as the field and is rotating inside (conceptually) three coils of wire spaced 120 degrees apart. As the magnetic poles pass by the each coil voltages that are 120 degrees apart are induced, this is known as the open circuit voltage of the generator.
Things get more complicated once the generator is attached (synchronized) to the grid and load is applied (the actual process of doing so is another topic). Once that happens the three coils of the stator create a rotating magnetic field of their own which in turn interacts with the rotating magnetic field of the rotor. It is the relative strength and balance of these fields (called the air gap flux) that determines the reactive power (VARs) output of the generator.
The real power portion (Watts) comes from the interaction of the mechanical power applied to the shaft which is turning the rotor. It is doing work to keep that magnetic field rotating, and that work is being transferred magnetically to the grid through the stator coils' magnetic field from the grid.
For a huge synchronous generator, there a system referred to as excitation system. This excitation system supplies a dc current to the generator rotor winding through brush and slip ring arrangement. This dc current sets up a magnetic field which by reason of a turbine turning the rotor of the generator through couplings, rotating magnetic field is created.
This field is rotating at the same speed as the turbine shaft driving the generator. These magnetic fields through the air gap links with the stator winding and an EMF is induced in the stator winding. The excitation current is increased until the generator attains the rated EMF.
From this point, if conditions of synchronism are met, the generator connects with the utility service. By the use of synchro scope, these conditions are not missed. For you to meet up with the load demanded from the generator, you increase your air, fuel and feed water flow for a thermal power station or increase the water head for hydro station. These make the generator meet up with the load that is demanded of it by increasing its torque at constant speed.