Why is swimming a good fitness exercise?

Many reasons. It is aerobic (if done at the correct intensity), it is water cooled (good especially in hot climates), and it is much easier on the joints (no pounding on the ground). However, if you can't swim or hate to swim, find something you will do. The best exercise is the one you will do.
Works whole body. Swimming is a terrific full body exercise. It can get your heart rate way up and is great for the cardiovascular system. Moving through water provides some resistence and helps increase strength. You stretch while you're swimming so you increase flexibility. Best of all, there's no impact, so people with back/neck/knee etc. Problems, people who can't do other aerobic things, can still do it.
Low impact. A major benefit of swimming is that it is low impact. The only common injury is shoulder type, and this is usually overuse and strain with competitive people. The water supports you and the strokes help stretch and elongate your muscles. You use your whole body in many ways. It will not help you lose weight like other exercises, because the water keeps you cool, and the legs are not as taxed.