There's an awful lot of room for water molecules in the air. Any that happen to fly up in the air are very unlikely to return once they wander a bit away from the starting place. Unless there are already a lot of water molecules in the air, the flow will be one way, from the liquid to the gas. In order to keep there from being net evaporation, there must be a flow back from the air to the water.
The concentration of water molecules in the air is often given by the relative humidity. At 100% relative humidity, the flow of molecules from a cup of water into the air will just balance the flow from the air to the cup, if theyíre both at the same temperature. Itís true that the rate at which the water molecules leaves goes up quickly as the water gets hotter and the rate at which the molecules returns only goes up a little as the air gets hotter. So the evaporation is much faster when the water is hotter, and 100 % relative humidity means a higher concentration of water in the air when itís hot than when itís cold. So thatís probably where the impression arises that water shouldnít evaporate when itís cold.
One interesting case of cold water evaporating occurs in your freezer. Ice cubes directly evaporate, although we use the word "sublimate" instead. Even though ice is a solid, its molecules are rattling around and occasionally fly off the surface, just as they do in a liquid.
All the evaporation occurs at the surface, so thatís a big reason why the rate depends on the surface area. Itís true that the evaporation cools the water, which would reduce the evaporation rate if no heat flowed in. As you say, heat can flow into the exposed surface, but it can also flow in from the container. So I bet thatís a less important reason for the surface area to matter.
(republished on 07/24/06)