Why during the evaporative cooling process the heat is taken from the water inside the earthen pot and not the heat of air outside the earthen pot?

Answer 1

During evaporative cooling, the most energetic (hottest) molecules of water are the ones that break free of the intermolecular bonds and evaporate. When this happens, the total average energy of the water molecules decreases, thus lowering the temperature of the water. The heat must come from the water, otherwise the water would be heating up, not cooling down.

The water also does absorb heat from the air outside the pot, this heat is what causes the water to continuously evaporate in an open system instead of cooling to the equilibrium point (also called the wet bulb temperature) and then sitting there for eternity.