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.