When you strike the drum it'll create a wave of sound inside the drum. Normally this wave starts bouncing around inside the drum, back and forth, giving a sort of lasting ringing a few moments after you hit it. That's not usually want people want, they want the sharp WUMP of the drum, and then silence.

So the pillow as an a "damper". To damp something is science-speak for "reducing the intensity" of something. In this case, it's reducing the intensity of the wave so that it dies quicker.

Not to be confused to dampen something, which means to make something slightly wet.