By doing your best. A teacher brings out the best in his students by giving his best in the classroom. A salesperson brings out the best in his clients by giving the best possible service and attention. A manager brings out the best in her employees by setting and example of diligent, consistent work. And it works in every direction.
Employees bring out the best in their employers by doing their best possible work.
Students bring out the best in their teachers when they are fully committed to learning. The people around you are a reflection of you. If you're focused, responsible and full of energy you'll find that the people around you will be the same way. People naturally tend to relate with you on your own wavelength. Your attitude is contagious.