In a growing child, there is a flat piece of cartilage near the end of long bones (like the forearm) called the growth plate. Eventually, when the child stops growing, these plates will form into bone. Until then, any damage to a growth plate can cause abnormal growth in the injured bone. This can lead to other problems by changing the way the bone lines up at the joint.
Your doctor is right to suggest your son not play the rest of the season. The injury must be treated until the symptoms are gone and full forearm and wrist motion is possible. Anything that causes pain and swelling worsen should be avoided.