These images show up mainly in lines 412-415, but they're pretty key (pardon the pun) to understanding the huge beef that Eliot has with modern people. Basically, what most of his anger boils down to is people's selfishness. In this sense, people live inside the private prisons of their own self-interest and ego. With nothing to live for outside themselves, these people spend their entire lives trying to fill the hole created by their lack of spirit or compassion, and they often do so by buying a bunch of stuff or taking advantage of other people, young man carbuncular-style. In either case, Eliot thinks people need to change the way they behave; but they won't be able to do this until they change the ways they think and feel about the world.
Lines 413-417: When he writes that "We think of the key, each in his prison" (414), Eliot means that each of us is trapped—either willingly or unwillingly—in the prison of our own selfishness and self-interest. He refers to this as a prison because he really believes that deep down, people's selfishness makes them wish for deeper connections with other people. Modern people have forgotten how to make these sorts of connections, though, because there's no unifying culture to bring us all together. So we all just continue doing our own thing and feeling lonely, assuming there's nothing we can do about it. Oh well.