Without beating around the bush, the major reason why towels (especially paper towels) absorb the water spills is that there are empty spaces between the cellulose fibers. The water easily fills up these empty spaces when it comes in contact with the towel. When paper towels are closely checked, they are seen to be made of ground-up plant material.
Taking a better look at it through the microscope, you will discover a web of tiny plant fibers that it is made of. When you magnify it closer, you will see that the fibers are made of sugar molecules that are connected together on a long chain. This chain of sugar molecules is called cellulose.
Another thing you should understand is that water easily gets attracted to cellulose. Thus, the cellulose attracts water and the water gets stuck to it. Also, there are some spaces between these fibers, and what happens is that, as water is being attracted to the cellulose, it also fill up these spaces.