Why would a boxer, for example, train jumping rope or doing anything that is not the same type of movement that happens in a fight? Can you just train by performing the activity you want to improve? Like a runner training exclusively through running, swimmer by swimming and so on.
There's more to training than just performing the movements of a specific sport. In the example you provided of a boxer, jumping rope will improve cardio vascular fitness and endurance. Think of how much power a boxer must generate and absorb while in the ring. Add a time factor of 8 to 12 rounds and you begin to realize that having sufficient endurance and lasting power in the later rounds may be the deciding factor to winning or losing a bout. Cross training is an effective method to improving overall fitness for specific sports. That's why athletes will tend to do more than just the movements of their sports.
Overall muscle development and training is very important for any athlete.
Using the example of a boxer like you provided, it is certainly most advantageous for them to practice hands on boxing, but this should be supplemented with other exercises. This is for a few reasons. First, a boxer needs training overall. Jump rope provides training with speed and cardio, both of which are useful to a fighter. Second, the body adapts to training very well. Doing the same training (in this case hands on boxing) will eventually become less efficient. This is why supplementing with other exercises is helpful.
Take powerlifters for example. You can absolutely bet that a powerlifter will be practicing his/her squats often. However he/she knows that in order to be better at squats, they should also perform some supplementary movements to focus on the specific muscles which are needed during a squat.