Most (maybe all?) large grid-scale generators use electromagnets to generate the electromagnetic field they use to generate electricity. These magnets require electricity to generate that field, so you need a small generator to kickstart your big generator's magnets in order to start producing power. There are other concerns, too; depending on the nature of the plant, there may be some other machinery that requires electricity before the plant can operate. It doesn't take much startup energy to open the gate on a hydroelectric dam, but I don't think anyone is shoveling enough coal to cold-start a coal plant without a conveyer belt.
If I had to guess, I'd assume that the generators on your destroyer were gas turbine generators that use some kind of motor as part of their startup process to get the turbine spinning. It's entirely possible that there was an electric motor in there to facilitate the generator's "black-start," but it may have been powered by a battery rather than a smaller generator.
Steam turbine generators (this was back in the 1970s, the ship was decommissioned in 2003). No motor to start turning the turbines, just apply steam. The main engines were also steam turbines, and likewise started just by opening the throttle valves--one for forward, another for backing. We did have a jacking gear powered by an electric motor, but it was only used to prevent warping as the engine cooled down when we went cold iron. And of course the boilers ran on fuel oil, not coal--coal went out on naval ships in the early 20th century.
As for how the generators' fields were started, now that you mention it I'm not sure. We did have emergency diesel generators (and of course shore power when we were pier-side), so maybe those supplied electricity to jump-start the generators. But they were 750 kw generators (upgraded in 1974 from 500 kw generators), so I don't imagine batteries would have sufficed.