What effect is described in the above story? Why would such an extreme reaction help reader understand what is happening in the story. The...

The passage that you are referring to is the escape by Mr. Nuttel from the Sappleton home at lightening speed.  He is running away from, what he thinks, are the ghosts of Mrs. Sappleton's husband and brothers who have just risen from the dead out of the bog.

It is amusing to think that Vera, has so altered Frampton Nuttel's perception of reality, that he believes her story so thoroughly, that he is terrified beyond his wits when he sees the three men approaching the open window. He thinks they are ghosts. 

He not only gets up and runs for his life, he leaves behind a perception for Mrs. Sappleton that he is rude and a bit crazy.  Running out of the Sappleton's home provides an opening for Vera to tell another tall tale, this time about how frightened Mr. Nuttel is of dogs. 

Saki is telling the reader that perception is reality.  He is telling the reader that, if an individual is convincing enough in her conveyance of a story or situation, that the story can shape or alter the perception of the listener, thereby creating a new version of reality.  Such as in Frampton Nuttel's world, where ghosts rise from the swamp to return home on the anniversary of their deaths.