I love playing with the Van deGraaf Generator. It feels a bit like magic. The only bit of the explanation missing from the video is why the tissue paper that jumps back and forth between the large aluminum dome and the smaller earthing dome, is initially attracted to the charged dome. The tissue paper should not have had any initial charge so there should not have been an initial force to make it move.
This is a case of charging by induction. The tissue paper is in an electric field so any charges that can move will do so in accordance with the force on each as determined by the electric field at the place where they are. This means that if you look at each little bit of the paper there is slightly more positive charge nearer to the Van derGrraff Generator and slightly more negative charge further away. Opposites attract and like repels but because the opposites are slightly closer than the likes there is a net force accelerating the paper towards the large dome. The rest is history.