Answer:
When water is warm it evaporates much quicker than when water is cool. Therefore, the heat from the sun helps to dry clothes quickly.
Wind - Another drying factor is the wind. If it is a windy day the water is evaporated more quickly due to the air moving / passing through the clothes.
Try this outside experiment on a windy day and compare your results to doing this experiment on a sunny day.
Be sure to do a fair test using the same items and same method.
So, to sum it up, why do clothes dry in the sun?
Wet clothes dry in sunshine because the water that makes them wet evaporates. This means that the water turns into a gas and becomes part of the air. Evaporation happens more quickly when water is warm, and this is why a warm day is better than a cool day for drying clothes. Evaporation happens more quickly if the air is moving as well - so windy days are good days to help dry your clothes quicker.