An eccentric exercise is one in which the muscle elongates while under tension, due to an opposing force being greater than the force generated by the muscles. A good example of this would be to look at how the triceps muscles operates while performing a bicep curl. The bicep muscle produces the greatest force, but the triceps must operate to stabilize and control the weight. So as the arm moves from straight to bent, the triceps muscle must contract but is generally elongating. Since muscles are approximately 10% stronger during eccentric contractions, weight lifters often follow the initial action with a "negative" lift where they allow the weight to return to the starting position very slowly. In the case of our example, this means that the "negative" lift of allowing the arm to slowly straighten would cause the biceps muscle to be performing an eccentric contraction.
The point of an eccentric exercise in conjunction with plantar fasciitis is to utilize this type of muscle contraction to strengthen the plantar fascia and the muscles surrounding/associated with it. As you strengthen these muscles and the tendon, you help your body prevent future inflammation and pain from the problem of plantar fasciitis. You should wait until your initial inflammation has subsided before beginning these exercises. Once you begin the program, you will gradually strengthen the muscles and plantar fascia. Pairing this with other treatment and prevention methods like a home stretching program, orthotics, night splints, ice rollers, and other methods can help you find total relief from the pain associated with plantar fasciitis, and also prevent it from re-occurring.