They meet an ongoing market niche, or actually niches. In the US private schools enrolled more students than public schools until 1950, meeting the demand for education. Public schools tend to be slower, more political, and designed to meet a wide range of public need and economic development. They also tend to be larger than privates.

Depending on the level of regulation and accrediting, private schools are generally better, or worse, in quality (what students learn and the educational experience are the best measures of quality). So, in the US where every school, with a few exceptions for religious schools, needs to meet the same standard, private schools tend to be better, and tend to attract more academically able students. They also cost more because they get no public money subsidy.

Privates can pop up quickly, especially in technical and vocational areas, to meet a real market need that is not being met by publics. In developing nations private schools can get going quicker to meet the need for education.

Think of education as an industry meeting public need. If the public schools won’t meet a need, some other school will. There is also an interesting split between for-profit, and not-for-profit schools.