Why does the headless horsemen chase Ichabod Crane in "The Legend of Sleepy Hollow" ?
Most people believe to a certainity that the headless horseman is Brom Bones. That being the case, Brom knows very well the degree that Ichabod is scared of nearly everything. His objective is to keep Ichabod away from Katrina Van Tassel, so he hopes to scare him very badly and make him stay away.
It is speculated that Brom wanted to kill Ichabod, but I don't see anything in Brom's character to indicate that. Undoubtedly, since Ichabod is not seem in Sleepy Hollow again, it is likely that Ichabod learns that Brom had let other people in on the headless horseman stint and Ichabod is too humiliated to be seen in Sleepy Hollow again.
In any event, the chase is just the start of this whole sequence of events and speculations.