Last weekend I saw a pretty awesome demonstration of why some smaller birds go after larger birds. I was driving along the highway when I saw a very, very large bird circling about half a mile ahead. As I got closer I saw that it was a bald eagle and it was being dive-bombed by a crow.
Just as I drove past, the eagle swooped down and picked up a dead cat from the grassy highway division. So in some cases it's just a matter of competing for food. Other times it might involve chasing away a predator (some species of hawk eat songbirds, so those songbirds may try to gang up on the hawk).