The evap cooler is the 'radiator' that cools the return air....warm air when cooled drops the moisture. It runs down into a drip pan under the evap cooler.......the drip pan has drain holes so the collected water can drain away.
It flows down across roof and drips onto ground.
IF you don't see water running down from roof when A/C is ON the drain holes are probably clogged and you will get water dripping from ceiling grill......blower fan will carry it over into room air discharge.