Why does the headless horsemen chase Ichabod Crane in "The Legend of Sleepy Hollow" ?
The headless horseman appears as a specter who haunts the scene where Major André was killed and he chases Ichabod Crane.
In "The Legend of Sleepy Hollow," the headless horseman confronts Ichabod Crane on his way home from the Van Tassel's party by the side of a bridge. He rides at the same pace as Crane as though he desires a race with him, a race similar to what Brom Bones has claimed to have had with this ghost. When the light sets the horseman into relief against the sky, a horrified Ichabod observes that this ghost is headless. As the frightened Ichabod realizes that the head is carried by the rider on the pommel of his saddle, Crane "rained a shower of kicks and blows upon old Gunpowder" and fearfully hurries toward home, but the specter's dark horse keeps pace with him.
When the two horses reach the road turning off to Sleepy Hollow, Gunpowder, who seems "possessed with a demon," plunges headlong down a hill to the left. The road that the old horse goes on leads to the famous bridge of the ghost story. Somehow, though, the girth of the saddle loosens, and Ichabod has to grab the horse's mane as the saddle falls off. An opening in the trees gives Ichabod hope as he perceives that the church bridge is near. "If I can but reach the bridge, I am safe," thinks Ichabod. Old Gunpowder is given a hard kick in the ribs, and he jumps upon the resounding planks of the bridge. When Crane turns to look behind him, he sees the goblin standing in the stirrups of his saddle as he hurls his head at the frightened schoolmaster. This "horrible missile" hits Crane, sending him headlong into the dirt as Gunpowder and the black steed speed past him.
The next day the old horse is discovered without his saddle, quietly grazing at his master's gate. His saddle is gone, and the reins are under his feet. The children have come to school, but there is no schoolmaster to be found. When a search is conducted, only the hat of Ichabod Crane is found by the stream near the bridge; close beside this hat is a smashed pumpkin. Later on after the people of the town research the stories of the Galloping Hessian, they conclude that Ichabod has been carried off by this villainous specter.
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.