How evaporation cools skin

Evaporation cools skin because the molecules that leave are, on average, the faster ones. The heat that paid for their escape is drawn from the wet skin they left, so the film and the surface under it end up cooler.

Sweat does not cool you by coming out of the pore already cold, and a bead that merely sits there is not yet doing the work. The cooling is the phase change, so wiping the moisture off before it evaporates also wipes off the mechanism.

The mechanism is more particular than the slogan. Molecules in the liquid share a range of speeds and pull on one another. A molecule at the surface gets free only if it has enough energy to break those pulls, and each departure removes more than the average share, which is latent heat leaving with the vapor.

Follow it one step further. Air moving over the skin carries that vapor away and lets more liquid follow. If the air next to the skin is already crowded with water vapor, molecules return almost as fast as they leave, and the net cooling shrinks even though the skin is still wet.

The case worth keeping in mind is specific. Step out of a pool into a dry breeze and wet skin can feel sharply cold. Step out into air that can take almost no more vapor and the same water sits on you with much less of that bite, because net evaporation is slow.

That case has a boundary. The account assumes the vapor can actually leave. A waterproof layer over wet skin, or a glove that traps moisture, stops the net evaporation, so the water and the heat both stay and the damp stops being a cooler.