Molecules in a liquid don't all have the same speed, why does a liquid cool down when it evaporates?

You're right. The molecules in a jar of liquid have a wide range

of different speeds. The average of all of their kinetic energies

is what we sense as the liquid's temperature.

Within the jar, there's conduction and convection going on

constantly. Tiny warm samples conduct heat to the tiny cool

samples next to them, so that the temperature stays uniform

throughout the jar.

When some of the liquid evaporates, it's the fastest molecules that

escaped from the surface and never returned to the jar. With them

went their little bit of kinetic energy, so the total kinetic energy of the

molecules that got left behind decreased slightly, and since it was

the most energetic ones that departed, the average KE of those that

remained also decreased. Once this new average gets uniformly

distributed throughout the jar, it'll show up as a decrease in temperature.