Vera's fabricated story convinces Mr. Nuttel that she is telling the truth for three main reasons. First, he doesn't personally or casually know the men involved in Vera's story, so there is no prior knowledge from which Mr. Nuttel can draw a conclusion that she is lying. In fact, Vera makes sure that he doesn't know the men before she tells the story by asking him, "Do you know many of the people round here?" Then she rephrases the question to make sure Mr. Nuttel doesn't know the family well by asking, "Then you know practically nothing about my aunt?" Once both questions are answered negatively, then Vera can tell her fictional tale without being discovered as a liar or a prankster.

Next, Vera is a great actress and convinces him through her body language and facial expressions that she actually believes that the men approaching the house are supernatural. The following passage describes her well-played acting skills:

"The child was staring out through the open window with dazed horror in her eyes. In a chill shock of nameless fear Framton swung round in his seat and looked in the same direction."

Since the passage says that Framton swings around to see what the child is looking at, it can then be inferred that Vera's acting skills are very convincing. 

Finally, the fabricated story is a believable one. In fact, Vera doesn't tell the story of an unlikely tragedy at all. It is entirely possible for men to become trapped in swamps while hunting and fall victims to a fatal accident.

"In crossing the moor to their favorite snipe-shooting ground, they were all three engulfed in a treacherous piece of bog. . . Their bodies were never recovered."

Therefore, Vera's story is believable because Mr. Nuttel doesn't know the men about whom the story is told, Vera is a convincing actress, and the tale of the tragedy is plausible. Mr. Nuttel proves Vera has convinced him when he runs out the door after seeing the men who he believes are ghosts coming towards the house.