In the story, Vera does indeed show hospitality towards Mr. Nuttel. Her methods are a little unusual, however, and she entertains herself at his expense.
As the story begins, we learn Vera, a young lady of fifteen, has been charged with keeping Mr. Nuttel company until her aunt, Mrs. Sappleton, appears to greet her guest. When she first meets him, Vera tries to put Mr. Nuttel at ease. She graciously assures Mr. Nuttel that Mrs. Sappleton "will be down presently." In the meantime, if Mr. Nuttel is so inclined, Vera offers to keep him company.
Mr. Nuttel appears to ignore Vera after the initial introductions. He seems focused on the reasons he is in Mrs. Sappleton's home. Accordingly, Mr. Nuttel's sister arranged for him to meet some of her friends during his vacation. She has written him "letters of introduction" to present to these friends so Mr. Nuttel can become acquainted with them, too. It seems Mr. Nuttel's sister wants to prevent her brother from shutting himself off from people and succumbing to depression while he is on vacation.
Ironically, despite her good intentions, Mr. Nuttel is ill at ease with Vera. He is so self-absorbed and preoccupied with his own imagined illnesses that he initially neglects to engage Vera in polite conversation. Vera, being the imaginative soul that she is, decides to concoct a story about how Mrs. Sappleton's husband and her two brothers met an untimely death three years ago while out snipe-hunting. Of course, the story is not true, but Vera manages to engage Mr. Nuttel's attention so thoroughly that he has no time to think about himself. This is how Vera extends her hospitality to Mr. Nuttel: she keeps him engaged until Mrs. Sappleton appears.