Why does a blackboard dry very quickly?

• ^{[Already said]} A blackboard is not porous, i.e. it actually never takes up much water from the sponge in the first place (and if you were to squeeze out more than a little, it would just run down to the bottom).
• ^{[Already said]} Yet the surface is hydrophilic, i.e. the water that does stay on the board forms a very thin film instead of droplets (as you'd get on a plastic or freshly wiped glass surface), and together with the slightly rough texture this makes for a large surface area to only a very small volume of water. This surface is where evaporation takes place; the larger, the better.
• The board is mounted vertically. That's the ideal configuration for convection: water vapour has a lower density than air, so close to the surface (which, because of the second point, quickly evaporates a lot of water into the air directly next to it) the air rises up, and because the entire surface is aligned in the same direction and air can efficiently stream along the surface from below (turbulence helps further), there's a steady supply of unsaturated air into which more water can evaporate unhindered.
• ^{[Already said]} The bulk of the board is usually metallic, i.e. it has good thermal conductivity. To the touch (which emits heat into the board), one perceices it therefore as cold, but to the evaporating water (which requires heat) the same property has a warming effect. That keeps the evaporation speed high, both directly and through preventing the reduced temperature from mitigating the convection-causing density reduction.

Although flat the blackboard has a texture so that when a damp cloth is rubbed across it the water adheres to the board and isolated droplets are not formed.
So you have a fairly uniform thin film of water across the board which is ideal for evaporation.

I think the blackboards absorb very little if any water and, being smooth and vertical, there is just not that much water there after they've been wiped.

Also, the blackboards I'm most familiar with are made of painted steel. Probably their good thermal conductivity helps keep them at room temperature and not cool much as the water evaporates.