Why does the temperature of a substance remain constant during a phase change (e.g., melting or boiling), even though heat is continuously supplied?

The temperature of a substance remains constant during a phase change because the heat energy supplied at that point is used to overcome the intermolecular forces of attraction between the particles, rather than increasing their kinetic energy. This energy is known as latent heat. 

For instance, during melting, latent heat of fusion breaks the bonds holding solid particles in fixed positions, allowing them to move as a liquid. During boiling, latent heat of vaporization separates liquid particles into the gaseous state. 

Until all of the substance has undergone the phase change, the temperature will not rise.