Most modern homeschoolers have multiple reasons for homeschooling. It isn't something where I could pick just one reason. So, I'll give you various reasons that apply to us--not necessarily in any order.
Safety--Too many bullies, violence and other crimes going on in and around schools.
Educational excellence--The schools in our state tend to be lousy. Even the ones near us that are well-rated are good only in comparison to the other lousy ones. With homeschooling, my children are already ahead, learning things that children several grades ahead of them aren't getting in the public schools. Because homeschooling can involve individualized lessons, parents can skip things that the children already know, combine lessons that have a lot of repetition in them, and don't have to teach to the level of other, lower level, students (as in school), homeschoolers tend to get a better education.
Morals--Schools often subvert students' morals. They teach poor values, often without realizing it. Such as when they stop students in the elementary grades from "tattling" because it is too time-consuming to deal with the actual problems in a large class--and then they can't understand why teens are reluctant to tell on classmates who are threatening to blow up the school or commit other violent crimes. Or they tell students that they must have a certain level of grades to pass, but the students quickly discover that they can pass by taking a couple of extra days of school (yes, that happens here, 2 days can make up for a year of no work)--so students learn that people in authority lie to them and that rules are meaningless and a tiny bit of work is all it takes to get by.
Religion--If you are part of a religious group that celebrates holidays that may be different from the majority, the school will excuse the absences but will also expect the work to be made up and the child will miss lessons that he'll be expected to know. So, in essence the child is punished for it.
Cost--Homeschooling is cheaper than private school.
Freedom--We control the curriculum choices, our daily and weekly schedules, materials needed for classes, etc. If we want to take a family vacation during the school year, we can without worrying about missed work.
Efficiency--Without all the wasted time of a classroom (time spent on announcements that don't apply to most, or explaining the material for the fifth time for the students who didn't listen previously, or doing busywork because the teacher needs to get a mountain of paperwork done or has to work with other students, or lining up to go to the bathroom or the cafeteria or wherever, or working on material that the student has already mastered but must be covered for the benefit of someone else, or waiting while the teacher deals with someone else's discipline issue, etc.), education can be done so much more efficiently. The child then has more time to play, develop hobbies, spend with family or friends, relax, etc.
Family--We enjoy being together as a family, seeing our children learn. It feels so great to see a seven year old reading books designed for 12 yr. olds when you know that you are the one responsible for that.
Health--There've been recent scares about H1N1 and MRSA and other serious illnesses. Schools urge kids to stay home if sick but then penalize parents who keep their children home (calling them in for meetings about possible truancy even when they've been producing documentation of illnesses). And others don't keep their children home and schools, especially ones with no soap or towels in the bathrooms, are great breeding grounds for passing on those germs. When homeschooling, sick children can stay home from field trips or whatever; they can even do some lessons if they are up to it or skip them entirely (and not have to worry about catching up on makeup work later) if they aren't up to doing lessons.
Experience--I was homeschooled for a portion of my childhood and I know how great an education it can be. I wanted that for my children. And my spouse is currently a teacher in the public schools (and I have been previously); we know too much about what goes on there to want our kids there if we can help it....