When clothes are hung / dried in the Sun or in shade in a room or outside in the open, there is air around the clothes. The temperature of the air is higher than that of water. There is heat energy falling on the clothes.
As the air is not fully saturated with water at that temperature, the water from the clothes takes heat from the clothes and the atmosphere (Sunlight and air), and gets evaporated. Water becomes vapour and mixes with the atmospheric gases like oxygen, nitrogen and carbon dioxide.
When it is shining or hot, the clothes get dried quickly. This is because the air is less saturated than when it is not hot, air is damp or after a rain.