I do not believe that you can truly find any evidence for this in the paragraph itself. However, if you read the rest of the story you can see why it is important for Saki to state this in the first paragraph.
The story as a whole is going to center on the fact that Nuttel believes the story that Vera weaves for him. If he had thought she was just some crazy teenager messing with him, the story would not have come out as it does. So it is important for us to understand why Nuttel believes the story. This is why Saki refers to Vera in this way. Saying she is self-possessed helps to convince us that she is mature and must be telling the truth. A mature young woman like that would not (we think and Nuttel thinks) play such a trick on her guest.
The description of Vera as "self-possessed," is all part of the illusion that Saki creates throughout the first part of the story about his character. With the name Vera, the latin word for truth, and her self-possession, or ability to control her feeings, behavior, etc., and her use of an open window--suggestive of candor and honesty--Framton Nuttel, as well as the reader, is more eaily deceived by the tall tale of the fifteen-year-old niece of Mrs. Stappleton.
In addition, Vera's confidence ironically lends the appearance of maturity to her. And, as a mature young adult, one would assume that she would have more consideration for her aunt, with whom she lives, than to dally with such a serious subject as the death of relatives who are yet alive. That is, one would not suspect a respectable young lady to tell such a trifling story as Vera does. But, as Saki, with "tongue-in-cheek," comments at the end, "Romance at short notice was her specialty."
Saki emphasizes that Vera is "self-possessed."
"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."
Here the child's voice lost its self-possessed note and became falteringly human.
Vera is calm, cool, poised, relaxed and sophisticated for her age. The main purpose for presenting the character this way is for the sake of contrast. When the three hunters appear outside the house, all heading towards the open window with their guns, Vera's apparent loss of her self-possession is all the more effective. It is also believable, because we know that Vera is not really as self-possessed as she is trying to appear.
The child was staring out through the open window with a dazed horror in her eyes.
This goggle-eyed, open-mouthed look of "dazed horror" convinces Framton that these are the very same three men who were supposedly killed three years ago.
"In crossing the moor to their favourite snipe-shooting ground they were all three engulfed in a treacherous piece of bog."
They are the same three men that Vera's rattle-brained aunt has supposedly been expecting to return for tea for all those years. It is the look of horror on the heretofore self-possessed young girl's face that communicates fear to Framton and makes him flee for his life.
Vera can talk about the men being engulfed in a bog without showing much emotion, because she was not there when it happened and claims to know only what her aunt told her.
"Poor dear aunt, she has often told me how they went out..."
But when the three hunters actually appear outside, Vera can act incredulous and horrified for Framton's benefit. The girl knew she was going to have to fake such a look when the time came. Maybe she had already practiced it in the mirror and was just waiting for the proper opportunity to arrive.