Reasons for homeschooling
Homeschooling is a way for parents to take on the primary responsibility for educating their children. By primary I mean as the sole guiding source of instruction. While in many cases, state or third party curricula is used, as well as shared resources or even online courses and enrollment in distance education schools. Nevertheless, there is in many cases limited oversight.
The main reasons for wanting to homeschool is based on religion, though the underlying themes are of children being taught the wrong things, and not the right things, and in general a lack of appropriate schooling including education and environment. There are other reasons such as children with special needs, but it comes down to the same thing, a form of inadequacy. There is a small subset of special needs based on the lifestyle of the children such as young actors and athletes.
Reasons against homeschooling
The practice of homeschooling is illegal in many countries, and only in the United States does it reach into the millions of children. While the homeschooling movement includes many different people with many different reasons for homeschooling, the vast majority is promulgated by various religious sects.
Nearly all countries have some kind of public school system. Some have no private schools whatsoever, such as Finland which has made it illegal to charge money for education, thereby removing the financial viability of nearly all private schools. Finland also has an excellent education system, one that ranks as one of the best in the world.