Actually, from the 7 to 8 week it is not really necessary for them to remain with the dam as much as it is important for them to remain with their littermates for the socializing and interaction of other dogs. At four to six weeks, their development has barely begun and the few weeks following are very important in the life of a puppy as he develops.
The Puppy Toddlers stage is (3 - 6 Weeks) During the Toddler period, puppies emerge on their own from the litter. During this time, puppies learn basic behavioral patterns specific to dogs. While playing, they practice different body postures, learning what the postures mean and how they affect their mother and litter mates. They learn what it is like to bite and be bitten, what barking and other vocalizations mean and how to make and use them to establish social relationships with other dogs. Such learning and activity tempers their own biting and vocalizing. From the age of five weeks, the mother teaches her puppies basic manners. They learn to be submissive to her leadership and what behaviors are acceptable. If necessary, she growls, snarls, or snaps at them as a form of discipline. When weaning the litter, for instance, the mother will discipline her puppies so that they will leave her alone and they are not properly weaned until 6 weeks. However, the mother disciplines them in a way that they clearly understand, after a few times of her discipline, the puppies will respond to and know her body language even if it is just a glare. This is when they learn to accept leadership and discipline in its early interactions with other dogs and if they do not training later on is more difficult!
Puppies that are removed from the dam and littermates too early tend to be nervous, more prone to barking and biting, and less responsive to discipline and tend to be aggressive with other dogs. Generally speaking, a puppy taken away from its mother and littermates before seven weeks of age, may not realize its full potential as a dog and companion. To maximize the mental and psychological development of puppies, they must remain in the nest with their mother and litter mates until seven weeks of age.
Socialization Period (7 - 12 Weeks) it is at this age that rapid learning occurs. At seven weeks, puppies can learn and what they learn will have a lasting impact. Everything the pup comes in contact with will make a lasting impression upon him, as it never will again. They are eager to and will learn whether they are taught or not and what he learns at this time is permanent! That is why it is so important to be aware of what you are exposing your pup to during this time. Removing him from the litter early can have a negative impact on him for the rest of his life. When a puppy is between three and seven weeks, he is learning how to be a dog. Sudden changes, such as new surroundings or the absence of his mother and siblings should be avoided. Sudden change contributes negatively on a growing pup. Puppies have also just begun to recognize the people and other animals in his life. He is learning from his litter mates that biting hurts and Mom is teaching them to accept her as the leader of the pack.
To answer your question "what does the ***** teach them" I would have to say from the time she whelps the litter it is all connected and has a domino effect whether she looses interest at 6 weeks or 4 weeks. But a key factor would be the litter mates playing an important role in each others developement!
And above all, it is the law in most states.