How does activated carbon bind the adsorbate?
Activated carbon works by enlarging the surface area of elemental carbon. Activated carbon can reach surface areas of several hundred square meters to several thousand square meters per gram. Also elemental carbon is fairly non reactive, non toxic and cheap.



As seen in the picture, activated carbon forms a micro structure with incredibly small pores, so that many molecules can be adsorbed (the forces involved decrease with a factor $1/r^{6}$ with distance). The adsorption itself is an application of the London dispersion force (one of the Van der Waals forces).

I don't believe chemisorption has anything to do with this.