Why does clay absorb water?

Because clay particles are extremely small, clay has a very high surface area.

That is to say:

If one cubic meter of dry sand and one cubic meter of dry clay were spread out on the ground (so that you could not see the surface) you could expect the clay to cover an area at least ten times that of what the sand would cover.

Because of the smaller particle size the air/ pore spaces between the particles of clay are much smaller (than that of sand).

For this reason water is held more tightly against the clay particles through cohesive forces and less drains away through gravity. If water were poured on dry sand because of the large pore spaces and lower cohesive forces more water would drain through due to gravity (although some would remain).