Why does the temperature of the water stay the same when it melts and boils?

The temperature stays the same when a solid changes to a liquid because energy is required to break the forces between particles of water therefore changing the state of matter and separating the particles away from each other.

When a liquid boils, the energy is needed by the particles to escape the surface of the liquid and boil. Instead of raising the temperature, the energy goes into the particles' kinetic energy store so it has enough speed to escape the surface of the liquid.