The author does not actually use the word "cool" in describing Vera. The first paragraph reads:
"My aunt will be down presently, Mr. Nuttel," said a very self-possessed young lady of fifteen; "in the meantime you must try and put up with me."
Saki uses the term "self-possessed" twice in describing Vera, and an acceptable synonym for that term would be "cool." The reader pictures the adolescent girl as poised, self-assured, dignified, and as not being in the habit of showing her feelings, especially to strangers like Framton Nuttel. Vera has been assigned the role of temporary substitute hostess and seems to be trying to act more polished than she actually feels.
The other place in which the author describes Vera as "self-possessed' comes shortly later with the following dialogue:
"Then you know practically nothing about my aunt?" pursued the self-possessed young lady.
Saki's reason for emphasizing Vera's poise, self-assurance, coolness, and self-possession, in both her physical demeanor and the tone of her voice, is to create a dramatic contrast when she is pretending to be horrified at seeing the three supposedly dead men returning towards the open window.
The aunt cries, "Here they are at last!" Nuttel can't see them at first because his back is to the window. He turns towards Vera "with a look intended to convey sympathetic comprehension." But he is shocked to see the change that has come over this self-possessed young girl.
The child was staring out through the open window with dazed horror in her eyes.
This change from self-possession to dazed horror help to convince Nuttel that the three men he sees when he looks in the same direction as the girl are ghosts. The reader still does not realize that Vera was setting Nuttel up for exactly the "chill shock" the poor man receives when he sees the "dead men" returning, all armed with guns, after being "engulfed in a treacherous piece of bog" for three long years. The reader does not realize the truth until Mrs. Sappleton's husband enters and the following paragraph explains Nuttel's hasty departure.
"Here we are, my dear," said the bearer of the white mackintosh, coming in through the window; fairly muddy, but most of it's dry. Who was that who bolted out as we came up?"
The story ends quickly, once the reader has understood that Nuttel was the victim of Vera's ingenious practical joke.