[MRM]

It's interior condensation related to the use of the bathroom/bath. No need to be worried. The warm moist air from all the uses of the bathroom is rising to the ceiling where it gets trapped, condenses, and the water and warmth support growth of something on the painted surface. If you check the humidity in those areas you'll find it is too high. The solution is more air transfer and lowering the humidity. You should install a bigger exhaust fan in the area and they should try to minimize using water in a way that encourages it to evaporate. That means shorter showers and not using as much hot water.

Your mold abatement is simple. Wipe the ceiling clean with a rag and cleaner, the way you'd clean anything else. For added comfort you can wipe it with a 5% solution of bleach. That's about a tablespoon of bleach in a half gallon or so of water. If you can smell the bleach it's enough.