At the freezing point, why is the vapor pressure of the liquid phase equal to the vapor pressure of the solid phase?
If the vapor pressure of the substance in its liquid phase would be different from the vapor pressure of its solid phase then there would be a transfer of mass from the phase with the higher vapor pressure to the phase with the lower vapor pressure.

This would involve releasing or consuming energy (heat of fusion) even in a closed system in thermal equilibrium, which is obviously a contradiction.