Why does water evaporate at room temperature?

Water evaporation is a function of temperature and pressure.

Boiling provides sufficient kinetic energy to evaporate all the water. In the absence of boiling, still water can evaporate. But this can happen only in an atmoshpere where relative humidity is lower than100% so that, air can allow some more moisture to saturate.

Water molecules are highly mobile and will have certain amount of kinetic energy, which provides sufficient vapor pressure to the molecules in the surface. This makes surface molecules to escape from the water. This can continue till the moisture in the air saturates.

The molecules of the water are always in motion when it is liquid. Some of those molecules manage to jump clear of the surface and are carried away as water vapor. 

Humidity, air pressure, heat and air movement are all factors that effect evaporation.