Temperature is a measure of the average kinetic energy of the particles in a substance. The boiling point is when the average particle has enough kinetic energy to break free of the intermolecular forces that hold it in the liquid. Because an average is being taken, temperatures below the boiling point can still allow the particles at the energetic end of the bell curve to escape.
When a puddle of water is left in the sun, the entire puddle will not reach boiling point, but the particles on the surface will absorb more heat than the average particle in the puddle and evaporate. Then the next layer of particles become the surface and this repeats until the puddle evaporates completely without ever raising the average energy to the boiling point.
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