In a liquid the molecules touch but can slide past one another, so a liquid flows and diffuses but can hardly be compressed. The fastest molecules at the surface escape: this is evaporation. It happens at any temperature and it cools the liquid left behind.
In a closed container, evaporation and condensation reach a balance. The pressure of the vapour at that point is the vapour pressure. It depends only on the liquid and its temperature.
A liquid boils when its vapour pressure equals the pressure outside. A change of state takes in or gives out heat while the temperature stays constant. That heat changes the forces between the molecules, not their speed.