Water evaporates because individual molecules acquire enough speed to break free from the attractive forces of the collection of molecules that form the liquid. At any given moment they move at a range of random speeds. Collisions make some of them faster and others slower. With rising temperature the average speed increases and evaporation is more intense. Once the surrounding air already contains water vapor, some of it rejoins the liquid and eventually an equilibrium is reached when no more net evaporation occurs.
Boiling happens when the liquid has enough energy to be forced to take a gaseous form. This transition cools it down, and more heat is required to keep it boiling.
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