Why does a small puddle of water evaporate at room temperature, even though the temperature is way below the boiling point of water?

To understand why the water evaporates, we need to know what we mean when we say something has a certain temperature. Temperature is proportional to the average kinetic energy of the microscopic molecules in the water. The important thing to note is the word average. There are molecules with a kinetic energy below and above the average kinetic energy. Some of these molecules have enough energy to escape the attraction from their neighbouring molecules and evaporate. Of course the rate of evaporation depends on other factors as well, such as how close to the surface of the water the molecule is. However the short answer is: the number of particles with a certain energy is a probability distribution determined by the temperature, as such some particles will have enough energy to escape and eventually the water will evaporate.