Many sports require largely single limb execution. Most of the action is one limb at a time while the other limb acts in a supporting manner, or is moving into the ready position. For example in basketball, the players jump off one foot (usually), shoot with one hand (using the other only for balance), and pass with one hand. Running happens one leg at a time. The same is true for soccer, football, rock climbing, golf, obstacle course racing, bowling, etc. So it makes sense to devote some training to getting stronger with single limb movements.
How exactly does training one limb at a time help enhance performance? Body-weight exercise, using one or both limbs, train not only the larger muscles, but the smaller supporting muscles to work as a unit to enhance overall performance and strength. You’re strengthening a whole kinetic chain. Ignoring the smaller muscles leads to weaknesses along the chain. Single limb training takes this one step further by training chains of muscle that more closely imitate common movements in many athletic endeavors. This leads to enhanced athletic performance, because all these kinetic chains are not just stronger, but have been trained to work together efficiently.
Now let’s talk about how the physique is shaped by this type of training. I mentioned the work of the smaller supporting muscles in single limb body-weight training. This leads to greater development of these smaller muscles, more on pace with the larger muscle groups. Because this work is a coordinated effort, there is more of an even development, rather than the major muscle groups standing out so much like you see in bodybuilders, for example. The physique is shaped differently, leading to a more capable look. The body looks ready to perform.
The third benefit I would like to talk about is the spill-over effect from single limb body-weight training and its application to other physical movements. For example any movement that can be done well with one limb, can be done well with two, at least from a strength point of view. If you can do a pull up with one arm, you can do a pull up with two, the reverse is not true. You also have a spill-over with unrelated skills where the movement patterns are similar. For example leaping, which is jumping off one leg, can help you get over a pile of bodies on the football field, over a parkour obstacle, or into someone’s face in a cage match.