The speaker has found what he believes to be a decent place to stay. Though the location is indifferent, the landlady lives away from the property and the price is reasonable, so he decides to give her a call. All that's now left is for the speaker to be honest and admit to the landlady that he's African. The suggestion is that racial prejudice is rife and that white people in post-war Britain will often refuse to rent out their properties to those they regard as racially inferior.
When the speaker tells the landlady that he's African, there's a long, pregnant silence over the phone. However, instead of coming right out and refusing to have anything more to do with the speaker, the landlady inquires as to the darkness of his skin. Eventually, the speaker describes himself as "West African sepia" but the landlady doesn't know what means.
After further assistance from the speaker, who tells her that it's like brunette, she responds by saying "That's dark, isn't it?" The implication is that the landlady doesn't want someone in her property whose skin she considers to be too dark. The speaker suggests meeting up so that the landlady can see his skin color for herself, but one gets the impression that she won't accept his offer.