How does Vera use the information she learns about Mr. Nuttel to her advantage?

Vera very cautiously and cunningly makes sure that Framton Nuttel is a stranger in this part of the English countryside and would not know anything that would contradict the totally fictitious story she plans to tell him. First she asks, "Do you know many of the people around here?" This is just the sort of question a hostess might ask a visitor like Framton, who is presenting a letter of introduction. 

"Hardly a soul," said Framton. "My sister was staying here, at the rectory, you know, some four years ago, and she gave me letters of introduction to some of the people here."

It should be noted that the author Saki has invented exactly the kind of character who will be a perfect victim for Vera's practical joke. Framton Nuttel is a stranger to the area, he is suffering from a nervous disorder, and his doctors have advised him to rest in the country. As he will tell Mrs. Sappleton shortly later:

"The doctors agree in ordering me complete rest, an absence of mental excitement, and avoidance of anything in the nature of violent physical exercise." 

Ironically, Vera joke will produce just the opposite results, including violent physical exercise when he flees the house and is last seen running for his life down a country road.

Vera continues fishing for information. 

"Then you know practically nothing about my aunt?" pursued the self-possessed young lady.

"Only her name and address," admitted the caller. 

This is enough to satisfy Vera. She doesn't have much time before her aunt will appear and take over as hostess. The girl quickly launches into her story about the three men who went bird-hunting three years ago and were sucked into a bog. If such a tragedy had really happened it would be known all over the region and Framton would have heard about it. Vera has to establish that he is clueless--and Saki has to establish the same thing for the benefit of his readers. Only a complete stranger would do for the role that Framton will play, and only a man who is suffering from what we nowadays would call a severe neurosis would react as spectacularly as Framton does when he sees three armed men approaching the open window in the deepening twilight.

One of the approaching men establishes that they are indeed the three hunters who were supposed to have died when they were sucked into the bog. He breaks into the song which Vera had told Framton he always sang when he returned home at teatime.

Noiselessly they neared the house, and then a hoarse young voice chanted out of the dusk: "I said, Bertie, why do you bound?"

And when Framton turns to look at Vera, he sees that: 

The child was staring out through the open window with a dazed horror in her eyes. 

That does it!

Framton grabbed wildly at his stick and hat; the hall door, the gravel drive, and the front gate were dimly noted stages in his headlong retreat. A cyclist coming along the road had to run into the hedge to avoid imminent collision.