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comment by ThurberMingus

I was homeschooled in Texas. I'll try to answer your questions.

    How exactly does homeschooling work? […] What does the federal law says about education?

Federal law says children must be educated. The details are up to the States. There are requirements stated must meet to get federal funding for schools, but homeschooling isn't funded and therefore the federal requirements don't apply.

State requirements vary. Here is a Wikipedia article that lists some of them. https://en.m.wikipedia.org/wiki/Homeschooling_in_the_United_States

    Do you have to apply somewhere?

Some states require notification, many don't. As far as I know, only Massachusetts and Rhode Island have an approval process to determine whether a child can be homeschooled, everywhere else just has notification paperwork, if anything. Texas, where I grew up, has no notification requirement.

    Are there literally no requirements to teach your kids?

Again, this varies. Several of the strict states require homeschooled kids to meet the same requirements and take the same yearly evaluations as public school students. This prevents parents from pretending to homeschool their kid, but of course ha the same problems as standardized testing does everywhere in the US.

Other states have much more lax requirements. For example, Texas requirements that "every student receive education in reading, mathematics, literature[etc, etc,], whether public or private." But they have a different rule which prevents the state from evaluating any private school that doesn't voluntarily join their accreditation process, and homeschools are unaccredited, and therefore not evaluated.

My parents did have us take some assessments near the end of elementary school times just to check on our progress, but it wasn't required, and they didn't have to submit the results to the state.

As far as college admissions: in the US, it largely depends on class rank, SAT/ACT test scores, and essays. All of the homeschools students I know that applied to colleges got in based on SAT or ACT scores and writing skills, and lack of class rank or grades from an accredited school went an issue.