Because the parents find the schools in their area unsatisfactory or not right for their children.
The groups aren't big like a school, there are a few children at different levels who are taught by parents who have specialist knowledge of maths, english, the sciences, history, music and art. Many kids thrive in a smaller group but still interact with others. As Rhiannon said there are online resources and they can be entered for exams externally.
I wouldn't have wanted to do it, neither would my husband, though we were very helpful. We enjoyed our work outside the home, our children went to good schools & were happy there. However I've known people who have been dissatisfied with local schools, home taught successfully up to GCSE level after which their children went to sixth form and did well.
I don't know why you are so anti. There isn't a one size fits all when it comes to children. We & our children were probably lucky.
Take your point about 'religious nutters' but home schoolers of that mindset are few and far between in this country, many of them choosing types of free school if they have the option. In any case home schoolers are inspected and regulated. If the children are doing well and happy, what's the problem?