Why does temperature remain constant when water is boiling?

This is because the external pressure is constant (at one atmosphere). If you increase the pressure, e.g. by using a pressure cooker, then the temperature goes up, or likewise if you reduce the pressure the temperature goes down.

Water boils when the chemical potential of the water is the same as the chemical potential of the steam. If we consider steam as an ideal gas then the chemical potential is controlled by the pressure and temperature.

If you start with just water at below 100ºC then the water evaporates, and the partial pressure of the water vapour increases until the chemical potential of the vapour and water match. At that point there is no net evaporation of the water.

However at 100ºC the partial pressure of the steam in equilibrium with the water rises to one atmosphere and it can't get any higher. So if you raise the temperature above 100ºC the water and vapour cannot be in equilibrium so the water boils continuously in a desperate but hopeless attempt to raise the steam pressure.

This is how pressure cookers raise the boiling point. At 100ºC the water boils, but in a pressure cooker it can raise the steam pressure to above an atmosphere so the water can remain in equilibrium with the steam above 100ºC.

The kinetic energy and temperature do increase, but these molecules are no longer liquid water; they detach and leave as vapor. If you measure the temperature of actively boiling water, the thermometer is affected by the hot bubbles around it and shows a temperature slightly hotter than the boiling temperature (it can be off by a degree or so, depending on the setup).

The energy is increased. However below the surface, water molecules are densly packed. So unless they can form a steam bubble the energy below the surface is immediately passed on to other molecules. Only when a steam bubble forms (or the molecule is on the surface) the energy is not immediately transfered on but the molecule can escape.

The energy necessary to become steam can be interpreted as the energy necessary for the water molecule to leave the liquid and not be bounced back in by the gas around the liquid. It is an equilibrium depending on the molecules mean movement distance without hitting another molecule or atom and transfering too much energy to that molecule so that it would become liquid again. The molecules mean free movement distance however is nothing else than the gas pressure. So with rising pressure it gets exceedingly difficult for the molecules to leave the liquid body of water because the chance of hitting another molecule in the air around it is getting higher, thus the boiling temperature rises.

When that molecule then becomes steam, it is no longer part of the liquid. When the molecule left the liquid it however took the excess energy with it. The excess energy is no longer in the liquid which remains at a constant temperature. The steam however does not necessarily do the same, depending on the conditions of the other gas around the liquid.

Also the reason why water boils with bubbles rising up from the bottom is mostly due to uneven energy distribution within the liquid. The heat source heats water molecules directly where it has contact to the liquid. Some molecules have so much more energy that even with the pressure the water exerts on them, they become steam while below the surface.

On a macroscopic level however, the liquid water temperature as a whole stays mostly constant, as the steam (having the higher energy) escapes the liquid leaving only the liquid behind at constant temperature.