The proof is Freyd's. I think it is fair to summarize the idea this way: Think of how every contractible space is a mere singleton, a terminal object, in the homotopy category, while a proper class of homotopically different spaces are quotients of contractible spaces. The terminal object has a proper class of different quotients. Indeed every object in the homotopy category has a proper class of different quotients, and so the category cannot be concrete.
In fact the result is very sensitive to which homotopy category you take. It is much more subtle than this version. But I think that is the core idea.