I'm American and I was home schooled by my parents out of Ideological incentives. I also know a lot of home schooled families. My experience is anecdotal but real and somewhat informed.
Home schooling has a number of outsize advantages that neither public nor private school has:
* Individualized attention and instruction especially in the early grades.
* An emphasis on self learning and self instruction in the later grades.
* It is more time efficient allowing young children more time for play.
* It encourages development of social skills by imitating adults not you peers. In my particular circle of home schooled children this resulted in being more comfortable holding a conversation with other adults and in general less problematic social behaviors with other children.
* In many of our cases including my own personal case it resulted in a better education than the public school was capable of providing. (I was on the path to failing if my parents had not intervened in this way.)
The trade off is that our family had to be single income for most of my childhood and thus very low income. We were poor though I didn't know it at the time. My mom and dad had to sacrifice quite a bit to make it work. Home schooling requires a large investment in money and time to be done efectively.
I by no means think that home schooling is for everyone. My wife and I had to choose a different route for our family due to a number of personal circustances. However for those who are prepared and capable of doing what it takes it can be a superior education for many children.
I also do not think that public school is evil. It's an incredible tool to empower people to better themselves. But like homeschooling it's effectiveness is a function of the community and society they are in being willing to make the tradeoffs and sacrifices needed to have an effective education system.
For those families who are in a society where that is not the case home schooling can provide a way out where there might not have been any other way.