How warm does water have to be to evaporate?

Answer 1

The process of evaporation of water requires heat - the latent heat of evaporation.
If you provide a source of heat to water it will boil. While boiling the heat supplied is carried away (as latent heat) in the vapor and the liquid remains at a constant temperature (boiling point).
Where evaporation occurs without a source of heat, the body of water is cooled (it loses heat) as the latent heat is carried away in the vapor (lick your finger and blow on it and you will feel your finger cool).
When water vapor condenses back into water, the latent heat it contains is given up as heat again. This warms the surface on which the condensation happens or the air around the drops of water formed. It is this heat energy that drives storm systems (such as tornadoes and hurricanes).

Answer 2

I'll take water as an example. It takes water molecules energy to pull away from the liquid water (against the attractive forces, since the molecules attract one another). This increases the molecules' potential energy; the energy is taken from the heat of the water, and the surrounding air, thus cooling them down.