Storytelling is important in the 1800s in the sense that it attempts to explain strange phenomena as well as to dispel more fanciful tales. The inhabitants of Sleepy Hollow, for example, are said to see and hear strange things, and some believe than a German doctor bewitched the place in its early days while others believe that an Indian chief--a wizard or medicine man--held powwows there before the Dutch settled the area. Stories are an attempt to find rational explanations of what they experience in the sleepy, shadowy, quiet place they live, but the story hints that Crane, at least, is silly for being frightened at the tiniest things that startle him, causing him to break into psalms. In the case of the Headless Horseman, a most incredible tale, this story implies that before we fear something, we should check our facts first.
This story speaks of Crane's inherent greed, and thus serves as a medium to condemn such baseness. Crane is described as a decent, well-educated man, but he courts Katrina only because he wants her father's wealth. He is not interested in settling down there, but in courting and winning the maiden so he can inherent some of the wealth so they can leave for parts unknown. The story shows how he is blinded by what he wants and is oblivious to the feelings of others, and the punishment he gets for it (being chased by the Headless Horseman and frightened out of town).
Finally, this story subtly condemns Crane's selfishness. Crane is already provided for by the inhabitants of the town (they give him room and board a week at a time in exchange for schooling their children), but he isn't satisfied with this. He seems to use his relationships with his schoolchildren to get extra food. Again, due to his selfish nature, we--the audience--watch with delight as Crane is pursued by the Headless Horseman.