This is an excellent question as it identifies how closely the setting is tied in with the action of this wonderful short story. Repeated references are made to the view in front of these two old ladies, and how stunning this view actually is. But I think it is important to remember that the view that they are focusing on is made up of the ruins of a former dead civilisation:
From the table at which they had beeen lunching two American ladies of ripe but well-cared-for middle age moved across the lofty terrace of the Roman restaurant and, leaning on its parapet, looked first at each other and then down on the outspread glories of the Palatine and the Forum, with the same expression of vague but benevolent approval.
Glorious this view may be, but we cannot escape the fact that they are looking at the wreckage of a dead civilisation. Of course, in a sense, the view is symbolic of what the women do. As they look at the past in front of them they also consider their own past, picking through the wreckage and bring to light verious secrets or relics. Reference to the Colliseum later on in the story likewise seems to add a gladiatorial element to the conversation between these two women. Thus the setting definitely serves to foreshadow what happens in this story.