Yesterday afternoon, my mother took me up to London to see the dentist. 

The exact setting of the story, as it happens according to what the girl explains, is right outside the café that the girl and her mother visit after going to the dentist. It is around six o'clock by the time they are done in the café, and they needed a taxi to take them home. It is raining, and the girl and her mother are standing on the pavement, waiting for the taxi under the rain. It is this specific setting that prompts a man who is nearby, and holding an umbrella, to go to the girl and her mother. He is a "well-dressed" man, also described by the narrator as "well-mannered," small, and around seventy years old.

According to the story, the man approaches the mother carefully, and politely asks if he could trade his umbrella, worth 20 pounds, for a one pound note. He claims that he lost his wallet but he needs a taxi ride home. His knees are weak, due to old age, and the one pound note would cover his cab fare. The girl and her mother were "soaking to the skin" in the rain, so the trade actually would make sense. 

After trading the umbrella for the one-pound note, the girl and mother witness how the man takes off hurriedly down the street, showing no sign of old age or weakness, and ends up in a pub. At the pub, he pays for a "treble," or a triple amount, of whisky. After drinking it, "he lifted from the coat-rack one of the many wet umbrellas hanging there, and off he went." 

This is when the girl and the mother, who are still watching all of this unfold, realize that this is the man's trick to get a quick pound note for a drink, while stealing umbrellas from the pub to trade. The rain is important to the setting because, like the mother says, in order for him to be able to get away with this, 

I’ll bet he prays like mad for rainy days.