What is the purpose of the active image sensor cleaning on my Nikon D3000?

Dust on the image sensor is very annoying as it creates a black blob on your images. Modern cameras have sensor cleaning systems which vibrate the image sensor to knock off dust particles. It appears to work quite well. With this system, you shouldn't need to clean the sensor manually unless you get a really nasty dust mote on your sensor. If the cleaning system can't remove a dust speck, take the camera to a professional service.

Boiler plate warning: You want to avoid cleaning the sensor yourself as you can damage the sensor if not done properly.

When changing lenses, use good practices to avoid getting dust into the mirror/sensor chamber. There's lots of suggestions on the web on how to do this. The cleaning system will do the rest.

Yes, the camera is cleaning the senor itself. No, do not clean the sensor manually. It may never need cleaning. When necessary, have a professional clean it.