It's about the water, not the air. Water, and other substances, have a "vapor pressure": a pressure at which molecules leaving the surface are in dynamic equilibrium with vapor molecules reattaching to the surface. If you pull a vacuum above liquid water, the vacuum will be repopulated with water molecules until the vapor pressure is reached. The equilibrium vapor pressure depends on the temperature at the boundary of the liquid.
Solid water also has nonzero vapor pressure; this is the reason for "sublimation" and also the reason why old ice in your freezer has a different texture than fresh ice.
Boiling takes place when the vapor pressure exceeds the pressure inside of the liquid: then it's energetically favorable to form bubbles.
It's not just water: molecules at the edge of a bound phase (solid or liquid) have a nonzero, temperature-dependent chance of escaping from the surface of any material. The volatile metals cesium and rubidium have particularly high pressures, and can make gases with interesting optical properties in a modest oven. Liquid mercury is frequently used as a barrier in vacuum systems because its vapor pressure is especially low, but not zero.