What change of state occurs when wet clothing on a washing line dries out?

It is connected with vapour pressure, which is determined by the amount of water in the atmosphere and humidity.

When the air is 100% saturated with water vapour there will be a dynamic equilibrium where as much water condenses out of the air onto the clothes, as evaporates from the clothes into the air. If humidity is less than 100%, then there will be more evaporation than condensation and the clothes will become drier.

If the humidity is very low, either because the air is hot and can hold more water, or because there is just very little moisture in it, then the evaporation will proceed more quickly. If there is a wind this will speed the drying up because 'fresh' air, with low humidity is constantly passing over the clothes.