Nancy, it is legal in pretty much all states to homeschool your kids. And yes, that means they don't go to public school, and the government allows it. I am in PA which is one of the more strict states when it comes to homeschooling. I have to fill out an avidavid every year including objectives for each subject per each child that is of mandatory school age (in PA that is 8), have an evaluator, keep a portfolio of all my boys do for school (basically a sampling of their work in each subject) have a letter from my evaluator that says they progressed in that year and if it is a testing year then have the test results. I then turn the portfolio in to the school district for them to look over.
Most people homeschool for religious reasons but a few just want to have more control over what is taught to their kids. Some don't want their child taught about evolution, or abortion, or other alternative lifestyles (please don't be offended anybody). We could send our kids to private school but that is costly and homeschooling is a cheaper alternative.
Kids get socializaton through church, clubs, organized sports, swimming lessons, arts and craft classes, co-ops, music lessons, choirs and the like. We homeschool for the reasons listed above in the 2nd paragraph.
The government can't tell us how to raise our kids. And in some states where the school system isn't very good kids can get a better education at home. Homeschooled kids get into college, trade schools and go onto acheive great things.
Some homeschoolers choose to put their kids into public or private school for High School while others will homeschool all the way through.
I hope I have answered some of your questions.
:7