Why does cold air hold less water vapor than warm air? Why is relative humidity greater at lower temperatures?

The atmosphere is partially made up of water vapor
molecules which can move rapidly or slowly, depending on the amount of heat present.
 The determination of the amount of heat in the system is, in other words, to state the
temperature of the system.  Warm air means that the molecules are moving faster; cold
air means that they are moving slower.  In warm air, the molecules with their fast
motion are typically spread apart from each other.  In cold air, where there is less
heat present, the reverse is true -- the molecules have slower motion and are closer
together.   As a warm air system cools, the molecules slow down, get closer, and can
change state from gas to liquid -- in which case rain falls.  Cold air holds less water
vapor because it will tend to precipitate as rain, whereas warm air holds the energetic
water vapor molecules as gas.