Water evaporates because individual water molecules break free of the bonds that hold them all together as a liquid. While water evaporates more in heat, it is possible for it to evaporate in cold conditions.
All evaporation occurs at the surface of the water. Heat helps water molecules move at a faster rate. When they get moving fast enough, individual molecules have the ability to break free from the others that remain in liquid water to become gas. While increasing temperature generally helps, decreasing atmospheric pressure generally helps water evaporate, too. Decreasing pressure decreases the temperature that water needs to reach in order to evaporate. When boiling water at a high altitude, the water boils and evaporates at a lower temperature than water at sea level. This is because the pressure at the top of the mountain is much lower than at sea level.
While heat helps water evaporate, it is possible for the same process to occur in very cold conditions. In fact, water has the ability to change directly from ice to vapor. This process is not known as evaporation but sublimation. Even when in a solid state, water molecules still move around and occasionally fly away from the surface.