for heat to flow from hot to cold body there should be driving force which is temp difference..
but evaporation is case of mass transfer..in which driving force is difference between the vapor pressure and saturation pressure of water at that temp. As the difference increases evaporation increases and finally comes to an equilibrium state when two pressure become equal.
Clausius statement is that no heat can be transferred from cold to hot until work is done.
work is done to create the driving force( temp difference). here when air whose relative humidity is less then 100% comes,it swept away vapors with it from liquid surface thus lowering the vapor pressure and creating a driving force.
in the fig,cylinder is insulated from sides.when vapor pressure decreases,molecules at surface of water extract energy from neighboring molecules and become vapors,thus lowering the over all energy content of liquid water(temp).