Why does molecular potential energy increase with constant temperature?

The heat energy transferred to the substance undergoing a first order phase transition is used to break the molecular bonds of the substance characteristic of that phase. For example, when water ice changes to liquid water, one must provide the latent heat to break the bonds of the crystalline structure to the less tightly held together bonds of liquid water. Same story for liquid water to water vapor. You can see https://en.wikipedia.org/wiki/Latent_heat for more information.