You CAN leave your foals with their mom, if you choose to. Yes in the wild that is how it is done. In the wild the mare usually gets pregnant again, and eventually she will get tired of the growing baby trying to nurse and she will let him know by kicking or biting him or chasing him off. Eventually he will give up, although if a mom is willing to let him nurse some stubborn foals will keep trying for a loooooong time!
Most people who breed horses, want to wean them so that they can start working with the mare again, such as riding or showing and when they are weaned, this is easier, because they won't usually miss each other when they are separated. Or maybe the owner wants to start working more one on one with the foal, without mom around. Also some mares, when nursing have a hard time maintaining their weight or the owner may not want the mare to get too out of shape etc. Lots of reasons. When my mare foaled, I just gradually started separating them from each other for longer and longer times, they could see each other and visit with each other thru the fence or in stalls right next to each other, and then the mare's milk can dry up and I could start working with them individually. They never screamed or carried on at all, since they could see each other and visit. Then after several weeks when the milk was dried up I put them back together. It's good for the baby to be able to leave mom to be worked with and trained.
~He is the lord of all horses and has been my friend through many dangers - Shadowfax Lord of the Rings