There is a difference between dissolving and boiling.
Imaging that you pick out each water molecule one at a time and throw it into the air. When you have picked them all up like this, there is no more water. The water never reached the boiling temperature, but it still disappeared into the air.
This is what happens. You have probably heard about humidity levels in air, and they tell how much water air can contain as gas. If the max humidity level is not reached (dry air), the air will absorb water molecules, when possible by "picking out one molecule at a time" and freeing it from the liquid water.
When water is boiling, heat is added so that the water molecules are unable to stay together (they move around too "violently" and are ripped apart simply from being too energetic). When water is dissolved in air, it is diffusion because of the concentration difference that is powerful enough to rip molecules free (the air "wants" the water molecules more than the liquid water "is able to hold on to them").