The answer is that ballet is very athletic - it may look effortless & easy, but it's not! And so injuries can happen.
The jumping, turning, and other movements, combined with the sometimes un-natural turnout required, can put a great deal of strain on joints and ligaments/tendons (which are torn, not broken). The height which many dancers, especially the top men, can reach in jumps is stunning, and coming down from that height all it takes is to land bit off in order for an ankle or knee to give.
The strain is even more so for the woman when she must balance her weight on her toes on pointe and for the man in partnering where the woman can be lifted high, tossed and held is all sorts of difficult positions.
And yes, a lot of injuries are of repetitive nature and partly due to fatigue. Many hours of rehearsal a day, plus performances can wear away at the body and tired bodies/brains are more likely to take mis-steps. And old injuries sometimes don't get time to heal properly.
I think backs are the number one most injured body part for ballet dancers, followed by knees/hips/ankles. Shoulders can be a problem spot for men.