We homeschool for a number of reasons. The problem when telling anyone who asks exactly what those reasons are is that they are inevitably offended by one or more of them.
But here goes...
I want more of GOD in my children's lives than they will get in public school. Public schools are not friendly to the very idea of God, no matter whose, but they are particularly unfriendly to Christianity. This is my opinion, based on personal experience as a product of the public schools, and as a professional teaching in them. You don't have to agree. I am not attacking anyone else's decision to send their children to school. I don't think that makes you a bad parent. God gave you YOUR children to raise as you see fit. He gave me MINE to raise as I see fit. I'm not insulted by your decisions. If you ARE insulted by mine, perhaps you need to examine why you feel that way.
I want a better education for my children than they can get in public schools - by that, I mean one that is tailored to their particular learning styles, that does not waste years of time in pointless repetition and mindless regurgitation. It is not my experience that public schools teach children how to think. They are taught to accept what they are told as fact without testing or researching for themselves. Independent thought is only rarely encouraged. More often, it is actively discouraged. This is a function of the teacher:student ratio, not a slam on teachers' abilities or intent. No teacher will ever tell you that s/he can teach more to a larger group than to a smaller one. Certain behaviors and questions and rabbit trails simply cannot be tolerated in larger classes, or the learning objectives will not be met. My class has a 1:2 ratio. We have time for more than anyone in a 1:30 class ratio ever will.
About those learning objectives... every lesson needs them, but it is not the duty of the state to define them. My children are individuals, growing and learning at their own pace, not to be frustrated by goals either too easy or too difficult for their particular stage of development. My children are not to be defined by their age in years, but rather by their maturity of mind and body and spirit.
I want my children to have the opportunity to BE children - to go outside and play EVERY day, to develop their imaginations, to tell stories and act them out, to investigate whatever interests them at that moment (yes, even if it means scouring the neighborhood for worms and toads and snakes, Miss L).
I DON'T want my children's interests in ANYTHING to be discouraged. That means if my youngest daughter wants to be a herpetologist, she doesn't have 15 other little girls telling her "Only boys do that!" and "That's icky!)
There are other reasons, but those are the ones that come to mind right now...