God gave man stewardship over animals, and that includes using them for just purposes. Examples of just purposes that the Catechism of the Catholic Church points out are food, clothing, medical and scientific experimentation, and the work and leisure of man (CCC 2417–8). But man must not deliberately cause animals to suffer and die needlessly, because to do so would be contrary to man’s own dignity. Animals, properly speaking, do not have “rights” because they are not human. But man does have the human responsibility to treat them with reasonable care.
Does this exclude sport hunting? If a hunter were merely shooting an animal for the purpose of watching it suffer and die, yes. But the vast majority of hunters don’t do that. Some use the meat and skins of the animals. Others are helping to preserve the balance of nature by using carefully regulated licensing procedures to thin out animal overpopulation. Some hunt for sport, but the sport is in the tracking, gun skills, and trophy hunting, not in causing suffering and death to animals. All responsible hunters take care not to leave a wounded animal injured by a badly aimed shot to suffer; they make sure to track it down and end its suffering.
In short, the Church does not oppose sport hunting.