Why does the temperature of a substance remain constant during its melting point or boiling point?

Melting point: The melting point of a substance is the temperature at which it changes state from solid to liquid. At the melting point, solid and liquid states of the substance are in equilibrium; the melting point of a substance depends on pressure.

Boiling point: Temperature at which pressure exerted by the surroundings upon a liquid is equal to the pressure exerted by the vapor of the liquid under this condition, the addition of heat results in the transformation of the liquid into its vapor without raising the temperature. 

Now, When we start to supply the heat to a solid its temperature starts to increase up to the melting point of the solid then after the melting point of solids if we supply heat continuously to the solid but the temperature does not increase even after the continuous supply of heat until all solid melts. The extra heat supplied is used in the melting of ice which is called the latent heat of fusion.

Similarly, when a liquid starts boiling the temperature of the liquid does not change instead of a continuous supply of heat. The extra heat supplied to the boiling water is used in the vaporization of a liquid, which is known as the latent heat of vaporization. That’s why the temperature of a substance remains constant during its melting or boiling.