It's a play on the term "grass root movement". A grass root movement is one that starts with the people at the bottom, not at the top. So if a bunch of people got together (mostly independent of each other) to promote or protest something, that is a "grass roots" action.
Astroturf is basically fake grass, so "astroturfing" is something that is made to look like a "grass roots" movement but it isn't.