If you are a 9th grader, you might be able to have some insight into this because you would be about the same age that Vera is in the story. I think that her poise and self-confidence are really important in her being believed because they are perhaps not what you would expect of someone her age who is talking to a complete stranger.
Most of us who are adults do not really expect 15 year old strangers to act with poise when dealing with us. We expect them to seem more ill at ease when talking to some adult they've never met before. If a 15 year old acts the way Vera does, we automatically feel that they are more mature than their years. In such a case, we tend to believe them and have confidence in what they are saying.
So I think Vera is more believable because her actions are those of someone who is mature and trustworthy. And you would not expect someone mature and trustworthy to tell such a huge lie.
First of all, Vera's poise and self-confidence perceived by a man such as Framton Nuttel who himself lacks such things lends her a superiority. As superior in demeanor, she is thus somewhat intimidating to Nuttel; so, he would be afraid to challenge what she says. For instance, when Vera tells Nuttel that he must put up with her while he waits for Mrs. Stappleton,
Framton Nuttel endeavored to say the correct something that should duly flatter the niece of the moment without unduly discounting the aunt that was to come.
ceof Also, this self-confidence lends her a maturity beyond her years; this fact adds to the credibility factor as Nuttel may place her on a more equal plane with himself as well as giving more faith to her "testimony" that Mr. Stappleton and his two brothers-in-law went off for their day's shooting three years ago. Of course, the more mature idea of suggesting that the men left through an open window also suggests veracity [notice the similarity to her name] to Vera's tall tale. In addition, her poise enables Vera to fabricate with seriousness, a fact that affects Nuttel's credence, as well.
Saki emphasizes that Vera is poised and self-confident. He twice refers to her as “self-possessed,” which means in perfect control of herself—her speech, her gestures, her posture, everything. This description has a double purpose. It adds to her credibility because it makes her seem mature, but at the same time it contributes to the striking effect of the conclusion. Framton has his back to the French window. At first he does not see what is happening outside. He is looking in the opposite direction, at Vera and Mrs. Sappleton. We know that he is looking at them both and cannot see behind him because Saki writes: “…her eyes were constantly straying past him to the open window and the lawn beyond.” She sees the men approaching and says, “Here they are at last!” Framton turns to look at Vera with an expression “intended to convey sympathetic comprehension.” But “The child was staring out through the open window with dazed horror in her eyes.” Here Vera pretends to have lost her poise and “self-possession” in order to produce the effect she desires. At this point Framton “swung round in his seat and looked in the same direction.” It is only then that he sees the three figures walking across the lawn towards the window in the deepening twilight. It is because of Vera’s premeditated loss of her well established self-possession that Framton is convinced that these are ghosts approaching the house. So Vera's poise and self-confidence contribute to her being believed, and her loss of poise and self-confidence also contribute to her being believed.