You have the right idea. The cartridge translates the grooves on the physical surface of the record into electrical current. One end of the stylus rides the grooves and the other is attached to a magnet. That magnet sits between coils of copper wire. And as we all know (now you do!) a magnet moving next to copper wire induces current on the wire.
The tone arm is important in this picture because it needs to be relatively inert acoustically. If it amplifies or changes the vibrations that the needle picks up, then it will affect the sound.
