The key to this answer is Saki's ironic last sentence:
Romance at short notice was her specialty.
Vera, whose name comes from the Latin word for truth, takes the truth and romanticizes it cleverly into a tall-tale. But, before she does this, Vera is very shrewd. This "very self-possessed young lady of fifteen, asks Framton,
'Do you know many of the people round here?'
'Hardly a soul,' said Framton.
He made the last statement in a tone of disctinct regret.
When Vera ascertains that Framton is ignorant of the Stappleton family or anyone connected to him, she realizes that she can easily fabricate a story which the nervous and ill at ease Framton will accept in his apparent desire to know about people as his tone is one of "distinct regret."
As she weaves her tale of Mrs. Stappleton's "great tragedy," Vera pieces together the forthcoming events with the fictional events that she creates, centering them around the open window, an object that, with its openness, suggests honesty. To add to the effect of tragedy, Vera dramatically breaks off "with a little shudder" just as Mrs. Stappleton arrives, mentioning the forthcoming return of her husband and son.
Mrs. Stappleton's yawning disregard for Framton's feelings as he "announces" his condition and need seems to underscore Vera's suggestion that Mrs. Stappleton is delusionary. Then, when the men arrive and Mrs. Stappleton does just as Vera has predicted, Framton looks at Vera with "sympathetic comprehension." However, Vera dramaticizes the moment by staring "with dazed horror in her eyes," thus making it seem as if she has just seen ghosts, an action that frightens the already mentally excited Framton.
Clearly, Vera's penchant and practiced talent for the romantic--the exaggerated and disquieting events, the mysterious atmosphere, the tragic milieu--along with her understanding of her aunt's supercilious nature and the vulnerability of her listener, Mr. Framton, are what contribute "at short notice" to the realism of this narrative.
This is one of the great "surprise ending" stories of all time, in my opinion. The surprised depends on Vera's ability to make Nuttel believe that she is telling the truth. I think that she makes the story realistic by her use of little details.
As she tells Nuttel the story of the men who supposedly died while out hunting, she puts in lots of details that make the story more convincing. She talks about how that year had been extra rainy. She talks about the dog and the white coat that her uncle would carry.
In addition, she makes the story realistic by being a good actress. Most particularly, she seems to be emotionally affected by talking about the death of her uncle.
Here the child’s voice lost its self-possessed note and became falteringly human.