The phase transition is the change of the liquid water to gaseous water vapour. These two different states are called phases of water.
The reason that the temperature no longer increases is, that the water begins to boil and the phase transition from the liquid to the gaseous state requires heat (the enthalpy of vaporization or latent heat), so the heat drives the phase change instead of heating the liquid further. The amount of water in your pot gets less and less as it evaporates, so you can't add infinite amounts of heat without increasing the temperature. Once all the liquid is evaporated, the heat input will heat the vapour to higher temperatures.
You can't heat the liquid above its boiling point while keeping the pressure constant, because the liquid is no longer stable if the temperature gets higher (so if you heat an infinitesimal parcel to a higher temperature, it will just be vapour afterwards), at the same time the phase transition can't happen instantaneously, because you need to put energy in for the phase change to happen. (We'll ignore the effect of super.heating here, where under special conditions you can heat the liquid above the boiling point.)
This kind of phase transition, that shows latent heat, is called discontinuous (or first order) phase transition. There are phase transitions that do not involve latent heat, e.g. the change from a ferromagnet to a paramagnet at the Curie temperature. Those phase transitions are called continuous (or second order) phase transitions.