No, water doesn't only turn into a gas at 100 °C. Every liquid has a vapor pressure dependent on its temperature. 100 °C is only the special case where the vapor pressure equals atmospheric pressure. Once that point is exceeded, bubbles form in the liquid, which we call "boiling".
Evaporation happens without boiling because the vapor pressure is non-zero. This process is slower because there is less pressure "forcing" the water vapor into the air.
Since the process is bi-directional, it also matters how much water is already in the air. Think of each molecule on the surface of the water having probability of detaching from the liquid and diffusing into the air. The higher the vapor pressure relative to the ambient pressure, the higher this probability. However, water molecules in the air also have a probability of condensing.
When there are few water molecules in the air, more will evaporate into the air than the other way around, and the clothes will dry. If, however, the air is humid enough and the clothes cool enough, water molecules in the air actually have a higher probability of condensing onto the clothes than they do evaporating from the clothes. In that case the clothes will actually get more wet. This phenomenon is commonly called "dew".
In typical situations of clothes on clothesline on a sunny day, the equilibrium reached where the same number of water molecules evaporate from the clothes as condense on them is what you call "dry". Even "dry" clothes in typical conditions still contain significant moisture, but not enough for us to feel.