Why has warm air a higher water saturation than cold air? I've wondered about that for a while but all I read is commonly just "Warm/hot air has a higher energy so when it hits the surface of a liquid, it has a higher probability to transfer energy, resulting in water molecules leaving the liquid".
Though it makes sense, I don't think that is actually a reason. It only decribes the phase transition itself but doesn't explain why hot air can contain more water than cold air.
Simply: Releasing something doesn't mean you can contain it (?!).
 A: 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.
A: It is easier to think of this in terms of molecular motions. If the air is warmer, it will warm the surface temperature of a liquid in which it is in contact. This will increase the thermal energy of molecules in the liquid, increasing the probability that a molecule has enough energy to leave the liquid and join the vapour.
I am not clear why you think air "contains" the water? It doesn't. It only warms the surface to enable evaporation. Vapour is not "contained" (unless by a vessel). It consists of those molecules which have escaped containment. Vapour pressure is the partial pressure of vapour in the air, and not (at least to first approximation) altered by air pressure.
