They have to “boil” but not get the water to 100ºC. Water evaporates in the air at any temperature; it’s faster when the water is warm, and the air is warm and dry. Technically, that’s boiling, even if it’s not exactly like how your kettle does it.

Essentially, they find an equilibrium between the cold water coming in, warming in the sun, an increasing amount evaporating into the warm, damp chamber, and the remaining brackish water being cooled by the new water.