The usual explanation for photon's higher frequency in lower altitudes (higher gravity), when the photon is going downward towards a massive body, is that gravitational potential energy is converted into photon's energy which, by its turn, through $E=hf$ implies higher frequency. Because, since $c$ is constant, it doesn't make sense to turn that potential energy into kinetic energy.
That, I understand.
My question is, why is it affirmed that this implies that time runs slower in the lower altitude? I mean why is it such that this higher frequency is equivalent to slower running clocks?