Wet clothes kept near heater dry up because of -
Water in clothes evaporate.
Water from clothes get evaporated due to the heat from the heater and so they dry up. This is not a process of boiling as the temperature at which drying happens is less than 100∘ (boiling point of water). Same phenomenon takes place when wet clothes are kept under sunlight but the process is much slower as heat from sunlight is less intense.