Why do body builders work on certain muscle groups on a given day and then give them a rest period?

It is important to understand that training in the gym is catabolic, which means that it breaks down your body. Therefore, you do not grow in the gym. You grow during recovery, especially if you get proper rest and nutrition. That is anabolic.

Advanced bodybuilders often rest certain body parts while continuing to train other body parts. If you are a beginner, do not follow that example or you are liable to become overtrained. Instead, do a whole body workout. If it is of sufficient intensity, you will experience DOMS (delayed onset muscular soreness) the next day or two. Wait at least 24 hours after all DOMS has disappeared before training again. This will ensure systemic as well as localized recovery; you will not become overtrained, which can make you smaller, weaker, and sicker.