How does Vera’s narration in paragraph 14 help build suspense?

The key to answering this particular question lies in the way Saki's character, Vera, weaves her story: she hands out information only gradually, one piece at a time. This is an effect that can already be observed long before we actually reach the fourteenth paragraph. (In fact, I think the suspense present in paragraph 14 actively relies on the hooks that Saki has set up in the dialogue preceding it.)
Consider paragraph 10, where Vera creates a hook to capture Framton's imagination (while Saki, likewise, attempts to hook his readers'): in reference to her aunt, she tells Framton, "Her great tragedy happened just three years ago." If you notice, Vera does not really say anything. She merely alludes to some greater mystery. In so doing, however, she can continue building on Framton's own curiosity and expectations, dragging him further into her story.
From here, she can begin building her story, adding more details one piece at a time. In paragraph 12, she says, "You may wonder why we keep that window open on an October afternoon," which introduces only another layer to this mystery without resolving anything. Thus, the storytelling present in paragraph 14 actually builds upon hooks that were already set up earlier in the course of the dialogue, as to the yet unknown tragedy of her aunt and the mystery of the open window. These hooks entail a promise from Saki to his readers that these two questions (what was the tragedy and what does the open window have to do with it) will eventually be answered.
However, rather than answer immediately, he continues to build up additional details, creating a story within a story, as to the men who die within the bog, and the effect that their deaths have had on Vera's aunt. In the process, Saki is able to suspend that expected resolution, with the intention of drawing his readers deeper into this story-within-a-story he has provided.

Paragraph 14 of “The Open Window” builds suspense for the reader through a variety of ways. First, we must consider some background before this paragraph. The reader understands Vera to be “self-possessed,” meaning she is calm and confident. We have no reason to doubt her statements as being anything other than true. In addition, the reader knows that Mr. Nuttel has some difficulty with his nerves, giving us an indication that he may not be entirely stable.
In paragraph 14, Vera immediately calls attention to the window. The suspense begins when the reader now becomes aware of the importance of a window, which may not normally be of much concern. Next, the suspense builds when Vera says of her aunt, “three years ago to a day, her husband and her two young brothers went off for their day's shooting. They never came back.” The reader has been given details of a terrible event, and we know how the event ended, but the story is not over. This leads the reader to question what could happen next. As Vera continues the story, her voice loses “its self-possessed note and became falteringly human.” This means more questions for the reader, wondering what could cause this confident girl to falter. Finally, Vera mentions the window again multiple times, continually enforcing the feeling that something will soon happen regarding the window. Unfortunately for Mr. Nuttel, something does happen with that window.