How does homeschooling work?
As with all things education in the US, there are a lot of answers to that question, which vary by homeschoolers and by state.
In some states, homeschoolers are required to meet the same educational standards as public school students, including having to take state exams. In others, they are exempt from state testing. I believe that most states do require them to meet some standard, but that varies quite a bit.
Most homeschoolers use a purchased curriculum, as the Duggars do, which they buy from different sources. There are a lot of Christian-based curriculums, but there are others that are not Christian at all. Contrary to popular opinion, not all homeschoolers are religious. Some parents design their own curriculum.
Some homeschoolers pool their resources; different parents teach subjects they know to groups of homeschoolers. Some parents hire tutors for specific subjects. Some homeschooled kids do online classes. Some parents send their kids to public or private schools for specific subjects (usually science and/or math). Some send their kids to college for specific classes. And some just ignore certain subjects, like science, for the most part.
As you might expect, educational outcomes vary wildly. There are homeschoolers who get perfect SAT/ACT scores upon completing high school and homeschoolers who are functionally illiterate, and everything in between. Homeschoolers in my state are required to meet state education standards, so the homeschooled students I get in classes are generally right about where all the other students are. Some of them are behind the curve, but some of them are way ahead of it, too. I have learned to wait and see when a student says he is homeschooled; it could mean anything. Most homeschooled kids that I get are Christians (and that can mean anything, too), but I've also had kids who were pulled out schools and homeschooled because of bullying, chronic illness, mental illness, and lack of school support for giftedness.