why is the temperature constant when a pure substance melts and freezes?

Question

why is the temperature constant when a pure substance melts and freezes?

Answer 1

That's because all the energy gained or lost is used to undergo the phase transition. You are neither increasing nor decreasing the total kinetic energy of the system, but you certainly are changing its entropy. 

 

To put it another way, you only need the enthalpy of fusion to know how much energy is used during this process, while at any other point where no phase transition occurs you have to rely on the heat capacity of the substance, which is a derivative with respect to temperatureture of such substance.