Why is wet air less dense than dry air at the same temperature?

Explain why air of low humidity has a higher density than air of high humidity at the same temperature.

There is a lot of empty space between gas molecules. The molecules are very far apart, relative to their sizes, and they neither attract nor repel each other much. So distances between molecules aren't affected by the type of molecules in the sample. That implies that the number of molecules in a given container won't change, when you change only the type of gas filling the container. (This is one way to state Avogadro's Law).
Here is the same very small container, filled with dry air on the left, and wet air on the right. The drawings aren't to scale; the distance between molecules is much, much larger under normal conditions. Notice that in both pictures, there are 10 molecules in the container.
The molecular weights of O₂ (in red), N₂ (in blue) and H₂O are about 32, 28, and 18 atomic mass units, respectively. Substituting a water molecule for either an oxygen or a nitrogen molecule, then, will decrease the total mass of air in the container. Since the volume of the container is the same in both cases, the density of water-containing air must be less than the density of dry air.
Temperature and pressure affect the distance between molecules, so in making this comparison, we've assumed that both the dry and wet air are at identical temperatures and pressures.