How can I improve athletic performance through yoga?

Remove Restrictions
For every athletic movement, there are muscles that accelerate and also muscles that decelerate. Imagine you are throwing a baseball. The muscles in the front of your shoulders accelerate your arms to produce power for the throw. Towards the end of the throwing motion, the muscles in the back of the shoulders need to decelerate so your arm does not fly off of your body.
When the muscles that decelerate are inflexible, they will begin decelerating before you complete your full throwing motion. Your range for power production become less, so you cannot unleash your full power. To unleash the full power of your accelerating muscles, you also need to have unrestricted decelerating muscles.

Recruit More Muscle Fibers
Each muscle is an army of muscles fibers.
An army can get stronger in two ways. It can make the existing soldiers stronger. Or it can recruit more soldiers. Strength training is like the former - it makes your existing muscle fibers stronger. Yoga is like the latter - it helps you awaken dormant muscle fibers so you have more fibers to use.
Just because you know how to engage your biceps muscle does not mean you are able to utilize every single muscle fiber within that muscle. Many muscle fibers become dormant because we don’t need to use them in modern life. Yoga is a process of observation that will help you reconnect with these muscle fibers.
By making better use of what your body already have, you build an efficient body. You can become stronger without adding bulk to your muscles which slows you down and consumes more energy.

Prevent Injury
Most sports injuries occur from impact. That impact forces our muscles and connective tissues into an extreme position. In that situation, flexible muscles can stretch and come back to normal. Inflexible muscles will tear because it is unable to stretch.
Inflexible muscles versus flexible muscles is like brick versus bamboo. When hit hard, brick will shatter. But bamboo will bend and come back unharmed.