After he realizes that he has killed Eugie, Arnold goes into a state of complete shock. He acts automatically and continues the activity planned for the morning: picking peas. An unfortunate example of a child overwhelmed by grief to the point that he is unable to express any feelings, Arnold mechanically goes through the motions of picking peas---just half a bucket, which is his half of the task---before he returns home and tells his parents about Eugie. Arnold is completing the task assigned to him, doing his job, before he goes home.
Arnold's behavior is strange to everyone--his parents, the sheriff, the neighbors, even the reader--but psychiatrists indicate that his reaction to the sudden shock of his brother's death at his hands is not unusual for a child. His grief is extreme; he almost seems to have no awareness that Eugie is dead when he moves on to pick peas. In fact, he's probably in a state of denial, maybe even unconscious denial, for he simply cannot believe that his beloved older brother is dead.
He has become numb, unable to show grief. As the author indicates in her title, he is a "stone boy."
Arnold is in a state of shock, and picking peas is what he set out to do at first, so in order to make the day seem as normal as possible, he goes and picks peas.