I see this question asked a lot. Rodeo bronc busting or bronc riding has become highly controversial, in part due to its origins.
In the old days, cowboys didn't have much time to break a horse, so they would do so by the simple expedient of getting on, letting it buck them off, getting back on and repeating it until the horse realized it wasn't going to get them to give up. Like so much else from the old west, this became a competition. Cowboys would compete to see who could stay on a crazy bucking horse the longest.
Modern bronc riding is rather formalized. A ride lasts eight seconds, and it's split into two divisions - with or without a saddle. Cowboys who sit out the entire eight seconds are then judged by their style, most especially the fact that they're required to keep their spurs on the horse's shoulders. The horse is also scored.
This last is one of the reasons that a lot of people think bronc riding is inherently cruel. I'm going to go through the listed reasons one by one.
1. They're spurring the horses hard. In fact, the requirement to keep the legs forward and the spurs on is designed to...make it harder to stay on. There are strict rules about the kind of spurs that can be used and if you look at broncs, it's rare to see white marks on the shoulder or neck - when a horse is injured and scars, the fur often grows back white.
2. The horses only buck because they're annoyed into it by the flank strap. That's not true. These horses want to buck. Bucking is a natural reflex for a horse when a predator (and humans are predator) unexpectedly jumps on his or her back. We've mostly bred this reflex out of domestic horses. Broncs are chosen from the subset of horses that haven't lost it. The flank strap changes how they buck - it makes them kick out straighter, which prevents the horse from doing a twisting buck to the side which can be easier to sit, but can also result in a more dangerous fall. It does not touch their genitals or put pressure on their kidneys. Broncs are chosen for their desire to buck. If you put a flank strap on the average riding horse, it will buck a couple of times and possibly take off running, but it won't perform the wild, aggressive maneuvers of a true bronc. The flank strap also does signal to the bronc that it's time to do his or her job, just as a rein tells an animal to turn.
3. They use cattle prods. The use of cattle prods to get a horse out of the chute is very rare. It's banned by most rodeos, but is still occasionally used by some contractors. Once a horse has had to be prodded once, they are generally retired - a good bronc doesn't need that treatment, but one that has lost its edge might. A bronc that stops wanting to buck might be sent for breeding or even retrained as a riding horse. Many broncs, however, continue to perform into their twenties (although the best mares are retired for breeding long before that).
4. These are wild horses that hate humans and thus shouldn't be made to perform. Nope. Broncs are specially bred horses, which can command prices of five or even six figures. They are carefully chosen for both the desire to buck and the athletic conformation to do it well - just the same as any other sport horse.
5. Broncs tend to get injured a lot. This is also simply not true. The only person taking a real risk of injury in a saddle or bareback bronc event is the cowboy. Although broncs do get injured, the rate is lower than in, say, eventing and much lower than in racing. The bucking motion may put some strain on their back, but no more than being ridden regularly (broncs are not used that often because the more you use them, the more likely they are to get "broke" and stop bucking).
(Image source: Joekoz451 via Wikimedia Commons, public domain).
Note that there are bad operators in every aspect of the horse industry and there undoubtedly are stock contractors out there who treat their bucking stock badly - but the sport itself is not inherently cruel.