Imho, this process is driven not by energy considerations but by kinetic considerations. That should be why it naively seems weird that water absorbs heat from a cooler object and evaporates. Note: This is an explanation I came up with on-the-fly and have no references to back up with.
Since the earthen pot has small pores, water "flows" through those (probably by capillary action) to the outside surface of the pot. There, it encounters a relatively dry environment and it is kinetically encouraged to leave the surface of the pot and "evaporate". Presumably, while it is on the surface of the pot, it doesn't/can't interact much with the atmosphere and so it needs to take the necessary latent heat from the pot1,2. If that happens, it would help keep the earthen pot cool. Since the water evaporates, by Le Chatelier's principle, the reaction equilibrium gets a strong forward push and the water keeps evaporating.
[1] This seems like the weakest part of the argument. It would be interesting if someone could comment on this aspect.
[2] Since this is a surface effect, the cooling rate should now be propotional to the surface area of the pot. This would mean that smaller pots cool better than larger pots due to a better surface/volume ratio. This claims seems quite easily testable.