Why do cats purr?

Cats show happiness by purring. They may also purr when startled or upset.

I remember asking the same question a few years back! I learned that cats have special wiring that goes from the brain to the muscles in the voice box, which is able to vibrate the muscles so that they act as a valve for air moving past the voice box. The muscles work both while the cat inhales and exhales, which is why it seems like cats can purr continuously. The air passes through the valve, which opens and closes quickly to create the sound of purring!

The reason why cats purr is unknown, although the pleasant noise is associated with pleasure and comfort. However, the question "Why do cats purr?" is really not such a simple one, since they also purr when they are sick or afraid. In these cases, cats purr to comfort themselves.

Purring can also be an appeasement gesture. When animals do not want other animals or humans to hurt them in self-defense and want to show the potential predator that it is harmless and not worth attacking, the animal will show some kind of appeasement gesture. Among dogs, this takes the form of submissive movements, such as bowing the head. Cats purr to show the observer that the cat is really kitten-like and will not bite them (although it is not unheard of for cats to attack after they finish a round of purring).

In adult cats, purring is heard in domesticated cats when they are petted, and in feral cats during sexual or other social contexts (but I've heard on many ocassions stray cats purr when you pet them too).

Because they are happy or content with the situation. 

This is the sound they make when they are breast feeding. If they have a different pleasure in the future (even sometimes only imagine one), it reacts that pleasant experient from the past.

Cats purr to show their affection to its owner or just someone who is making them feel good.  They normally purr when they are being petted and it just feel good to them.  They want us to know that it feels good and to keep on doin it.  They are happy and they want us to know.  That is what i think cats purr.