Question

how does the desert cooler give cool air

Answer 1

When the fan of the desert cooler moves or rotates the water and the grass inside it gives cooling and the wind caused by the wind fan and the water droplets gets together and gives cooler air

Answer 2

The fan rotates at a high speed. That creates low pressure inside the cooler. Due to that the water evaporates into vapor. The vapor gets blown away. Water drops take heat from water inside the cooler.
So the cooler stays cool. The cool air from inside it is blown into the room.