How can potential energy increase without kinetic energy during phase change? From what I’ve come across the internet, the reason temperature remains constant during phase transitions is that the energy goes into increasing intermolecular potential energy instead of average kinetic energy per molecule (which temperature is a function of). But I still don’t understand how potential energy increases without any change in kinetic energy.
From the intermolecular potential energy curve, any mechanical energy added to a system results in an increase in both time averaged kinetic and potential energy. (For a simple harmonic oscillator, 50% of any energy added goes into potential energy). This explains why specific heat capacities of substances can vary significantly, depending on the proportion of energy that goes into potential energy.
By this logic, shouldn’t any change in potential energy be accompanied by a non zero change in kinetic energy?
 A: 
But I still don’t understand how potential energy increases without any change in kinetic energy.

It's because the phase change removes thermal energy from the system.  If this happens at the same rate that thermal energy is being added, then the total thermal energy (kinetic energy) is constant.

From the intermolecular potential energy curve, any mechanical energy added to a system results in an increase in both time averaged kinetic and potential energy.

How do you read that from the curve (and which are you referencing)?  The curve doesn't normally show KE.  We can raise the PE of species by separating them (moving them out of the PE well).  But that does not require that the KE change.
A: In a closed container with a liquid and its vapor inside, energy is required for a molecule to break away from the surface.  If a molecule at the surface gains extra energy in a collision with another molecule it can move into the vapor phase. Its energy in that phase will depend on how much of the energy from the collision remains after it has left the liquid.  The distribution of those energy is likely to be similar to the distribution of energies within the liquid.  At a given temperature there will be an equilibrium between the number of molecules leaving the liquid and the number which return.
