All those little water molecules are dashing about at random within the liquid. Some of them break free of the surface into the air. The higher the water temperature, the faster they're all moving, so more of them can become airborne, and so the water will evaporate more quickly.
You can turn saltwater into fresh water through evaporation, look up something called a "solar still", it's like a little inflatable pyramid and you fill it up with saltwater and slowly but surely the evaporation distills the water so you can drink it! Life rafts have them fitted as standard now.