A more microoscopic explanation is due to Maxwell-Boltzmann distribution.

First, you need to underestand temperature. The difference between cold and hot water is the average speed at which particles move, with hotter water’s particles moving faster.

But this is just the average speed, it turns out that particle’s speed can be se en as a random variable, and they follow Maxwell-Boltzmann distribution:

Maxwell-Boltzmann distribution

So you have a small proportion of particles that move very fast, even in cold water. If some of those particles get (or collide with other particles near) to the “layer” of water that is on contact with the air, they will have enough energy to escape water’s superficial tension, thus going into the air and out of the water body. The higher the average speed of the particles, the faster this process will go. Finally, the rate at which this process happens also depends on the energy required to be able to leave the water body, which depends on factors like air pressure.