I have 21st century auto insurance and i saw the commercial about how they extend your coverage to anyone you let drive your car. I was wondering how that worked, what are the requirements?

Because if that is the case then, one wouldn't need to buy any insurance for their children for example.

You have to pay for it. It's additional coverage.

You have to have coverage for every driver of the vehicle. That would include children. When a child reaches the legal age and obtains their license, the parent adds them to their policy and pays for it.