What is it mean when a father gives his rights cause the child is better off with moma. We are both early teens.

It means that you are listening to a myth, he can not just give up his rights, He can allow the mother to have full custody both physical and legal custody. But the father will retain rights to visit and he can always go back to court in latter years asking for a change in the custody

Giving up custody is a myth, he can not just give them up

Also if you have to go on welfare they will make you file for child support from the father.

It would help us help you if you explained the background here and the context of the question.

But generally it is a myth that a parent can give up their parental rights. They can cede custody, but they will still be responsible for supporting the child.

You as the father can give up your right to visitation but you are still legally responsible for for the child. As such there can be orders of support from the courts. You won't have any rights to decide legal issues either. So attending doctors or school will be beyond your parental authority. Support and medical expenses are what you will be responsible for.