What happens to water when wet clothes dry on a clothesline?

The water evaporates into the air. It becomes water vapour which rises up to form clouds.

The reason washing dries is because water evaporates from it. If a wet surface is in contact with the air, some molecules of water will leave the surface and go into the air, but at the same time, molecules of water vapour from the air will be coming into the surface. Eventually, it will reach some kind of equilibrium where the amount of water leaving the surface is the same as the amount coming in. We then say that the air is saturated with water, and once the air is saturated, no more [net] evaporation can take place. Now, if we look at the basic physics underlying this, we find that the amount of water that air can hold when it's saturated depends very strongly on temperature, and the warmer the air is, the more water it can hold. So, evaporation tends to proceed much more quickly when it's warmer than when it's cold. But even when it's quite cold, as long as the air isn't saturated, your washing will dry, but it may dry very, very slowly, and it may rain before it gets dry! In general, we don't hang washing out to dry in the Antarctic because it is so cold that things would take such a long time to dry. Maybe on a really nice sunny day in the middle of summer, you might get the tea towels dry, or something like that.