Water molecules can evaporate off a surface without needing to boil. This is because surface particles are sometimes energetic enough to overcome the surface tension, which is a technical term referring to the pull of other water molecules.
For these particles to overcome the surface tension, they need to be at something like T=600. (Not single molecules, but the set of molecules that are energetic enough to escape)
As the water particles fight their way against the pull of other water particles they lose so much energy that the temperature of the evaporated stuff becomes same as the temperature of the water.
What does atmospheric pressure have to do with this? Nothing, because there are huge holes between air molecules - or at least for a molecule those holes are huge.