Exactly why you get sore muscles after exercise is not yet fully understood, but most experts agree that two factors play an important role.
The first is the small damage to the muscle fibers that all forms of hard or excessive work involve. The damage is followed by light swelling and inflammation-like conditions that lead to stiff and sore muscles.
The other factor is free radicals, as we are bombarded with each day and that help break down the body. We have much larger amounts of these aggressive oxygen molecules in our body while we exercise and have high heart rate. The free radicals exacerbate the inflammation-like state of the body after exercise.