Inertia. If you push a piece of cloth, you'll create a wave of movement that will propagate on the tissue. In vacuum, there's no air to dissipate the energy in this movement, so the wave bounces back and forth on the tissue until the strain on it dampens the wave enough. Of course, there are other specific details about the interaction with the pole and whatnot, but that's basically it.

If you look at the video, the speed and amplitude of the movement at the only free tip of the flag, when they are putting the flag on the ground, is a clear sign of a vacuum. The little movement you see later is just the residual energy of that wave.

Since most tissue and all the other points can't move a lot due to proximity from the edges, only the tip seems to show any movement, but the energy is there, on the rest of the flag, bouncing back and forth with complex interference patterns.

And regarding to the specific claim on the video, if you take a closer look at the flag before the astronaut (not sure who it was) passes by, it was already waving, and I'm sure that if you had access to the entire clip you would be able to see it better.

__________________