The coaches screen shows you the areas your coaches have expertise in. The more knowledge a coach has, the better they can tell the potential of a fighter. It also increases the rate your fighters learn new skills (more knowledge, faster skill!)
The coach with the highest knowledge is automatically used when considering the above.
Clicking on a coach allows you to fire them or send them to training to raise their knowledge.
If a coach’s loyalty goes below 3, they will leave your gym.