Yes, he's right that bringing the poles of two magnets together puts the system in a state of higher potential energy. And, yes, you could use this as an explanation for "why" the magnets repel by invoking the principle of minimum energy. You can even show that this results in a force, as a gradient in the potential energy is mathematically equivalent to a conservative force. I do think, though, that you can give further justification for the principle of minimum energy than he gives in the video, as it follows from the second law of thermodynamics (see Wikipedia article). Regarding the exchange of virtual photons and using this to explain how the electromagnetic force arises: I would avoid this entirely.

One side nitpick though: I wouldn't say that the energy came from "the chemical bonds in the food [you ate]", but rather the formation of new bonds as you digest the food. Chemical bonds are states of lower potential energy, so breaking them in the sense of separating the constituent atoms requires energy. It's just that different bonds can have even lower potential energy and therefore release energy when they're formed.