Paper is made of plant fibers. In the process of making paper, the fibers overlap forming a massive network of tunnels in all directions throughout the paper. The chemical molecules making up the fibers are attractive to water water molecules. Adhesion is the name of the force of attraction between two unlike molecules.
FYI: I think of adhesive tape to remember this force. Adhesive tape is what I use to secure a bandage to skin. The tape molecules are attracted to skin molecules, thus the tape sticks.
Paper absorbs water because the paper molecules and water molecules attract, thus stick togethe