The benefits of yoga to athletes are pretty well documented now. They include:
Strength & Stability
Flexibility & Mobility
Endurance & Improved Breathing
Core - Yoga provides a workout that includes every muscle and all the joints. Yoga uses all muscle groups, including the small muscles and deep muscles.
It is a great aid in developing muscle strength, flexibility and balance, which all reduce the risk of injury and improve mental focus and breathing efficiency.
However I feel this should come with a word of warning:
A yogi who wants to boost his cardio with running should do hard yoga and soft running. On the other hand, a runner should do hard running and softer yoga.
This is true for any athlete. Athletes tend to be goal-oriented people with clear objectives: the main focus should be on their primary sport not on applying the same expectations to yoga.
Athletes will all have individual needs and probably a lot of expectations about what yoga should be for them. A lot of athletes like to go hard in everything they do. It can be hard to step back or say no to ego and competition.
So here is a second important warning about yoga and athletes:
Give them what they truly need not what they think they need
The great thing about yoga, and particularly Vinyassa, is that it is fairly modular and can be adapted to different needs. Common wisdom would suggest going hard with athletes and focus on a very dynamic flow in order to get a good sweat and workout.
Instead of strength, a lot of athletes actually need mobility and alignment. For them, yoga can be a way to step back and look at their core body mechanics in a meaningful way. Athletes can get pure strength from their sports and many other complimentary activities but very few other workouts can have such a deep impact on joint mobility and alignment and ultimately lead to a more functional body than yoga.
It might sound counter-intuitive, but it is a lot more productive not to teach power and flow yoga to athletes.
Instead power athletes would benefit from working on joint mobility and long holds that really emphasise functional challenges.