Answer:
Basically what it comes down to is the substance that is left behind during the phase change (say the liquid when it is changing to a vapor) has a lower internal energy than it had before the phase change started. To maintain the substance at its original temperature, an equivalent amount of energy must be added to make up for the loss of internal energy due to the phase change. (Conservation of Energy)The energy added in this case is heat, which in turn causes more molecules of the substance to change phases and lowers the internal energy & the process repeats. This is why the temperature remains constant as you add heat to a substance during a phase change. Once the phase change is completed the added heat will eventually cause the temperature to rise in the new phase of the substance.
Hope this explanation helps. It's kind of hard to explain without a basic knowledge of thermodynamics.