Can riding a home trainer improve fitness?

100% Yes. Even old school turbo trainers from the 90's could provide enough resistance to improve fitness. Former Australian pro' rider Mat Hayman won Paris Roubaix in 2016 training solely on Zwift - this followed a crash in which he broke his arm a few weeks earlier.

Besides the trainer, what else do I need when training at home?

There are a few things which definitely help when using a home trainer. An electric fan will help cooling and reduce the amount of fluid you lose through sweating. A trainer mat will help stop sweat from staining / marking the floor and reduce vibration / noise. Covering your bike with something to catch the sweat is also a good idea because sweat can corrode bike parts very quickly. If you are not using Zwift, or other training platform, you might benefit from music to listen to, or a film to watch on TV if you are up for a long session. If you are using Zwift, you will need a Zwift compatible trainer (check Zwift website), an Ant+ or Blutooth dongle, a computer, smartphone or Pad.

I need a very quiet smart trainer, so I don't disturb family, are there any?

Yes, some smart fluid trainers are very quiet, the Dare2Ride Fuego, for example only produces 52 Db at 30KPH, this is similar to the noise produced from a electric fan.

Is it hard to get used to riding rollers?

Riding rollers takes a little practice to get used to. It is much easier to start riding them in a door frame or next to a wall on one side - this makes it easier to start and stop - also there for any initial emergencies when you are starting.