Byron gets sick after hitting the bird for a couple of reasons. He had just eaten "a bag and a half of Swedish Cremes and some green apples", so he probably wasn't feeling too well to begin with. But the real reason he begins to vomit is because he is sickened by his own meanness. Byron always tries to act "bad" and "cool", but in reality he is a sensitive boy with a lot of goodness in him. When he sees that he has actually killed a living thing, the enormity of his act makes him sick to his stomach. Killing the bird is something "that...make(s) Byron hurt so much that he'(ll) have to think every day about how mean he is".
Byron reacts to his own vulnerability with characteristic anger, not wanting his concerned little brother to see through his tough-guy facade. He declares that it was "them apples got me sick" and pelts his Kenny with the leftovers so that Kenny will go away and not be witness to his remorse. In a show of touchingly innocent sensitivity and atonement, Byron then digs a grave for the bird he has killed, and marks it with a cross made of two Popsicle sticks (Chapter 6).