To me, there are a couple reasons why Framton Nuttel falls for Vera's story.
The first has to do with Vera. She is clearly a very good actress and that is not something you would expect from some 15 year old girl -- you wouldn't expect her to come up with a wild story and then tell it so seriously.
But the second has to do with Nuttel. We can see from what we are told of him that Nuttel is not the most sophisticated or the most social person around. His sister predicted that
you will bury yourself down there and not speak to a living soul, and your nerves will be worse than ever from moping.
If he is that sort of a person, he may not really be perceptive enough to realize when he is being tricked.
So I think it's a combination of Nuttel's gullibility and Vera's believability.