As particles are heated they will start to vibrate. The vibrations will increase as temperature increases.
As the water temperature increases the particles continue to gain vibrational energy until eventually they all have enough energy to leave the water phase and vaporize. Then the temperature change halts as all the particles have the energy to break apart the inter-molecular forces holding the water molecules together (Hydrogen bonding). Even though there is a heat source, the temperature of the water stays the same as the heat, latent heat of vaporization, is used to break the molecules away from each other to form a gas.
Read more: What_happens_to_the_particles_that_make_up_water_as_water_is_heated
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Molecules are always in motion. The molecules of a liquid at the boundary layer with air can move from the liquid into the air and not return. When enough molecules leave the liquid into the air and it is noticeable, we know it as evaporation. Eventually all the liquid can evaporate until there is nothing left.
When we add enough heat to a liquid, the added thermal energy excites the molecules converting the heat energy to kinetic energy and the molecules move faster. With the molecules moving faster, the ones on top move into the air at a faster rate increasing the speed at which the liquid disappears. We refer to this faster evaporation as boiling.