Why does a wet khus-khus screen hung at door keep the room cool?

This effect is due to 'evaporative cooling'. It is a very useful phenomena, and one may achieve a cooling of several degrees centigrade. In some hot countries, an open chimney is lined with damp cloth, and this cools the air in the chimney, which then falls because it is cooler and denser, and provides useful cooling to a quite large area.

When on motor camping trips, we used to hang our butter and milk in a wet cloth bag in the car, and this kept it usefully cool, and at no real cost.