Most schools are either private schools, which parents pay for, or public schools. Public schools are supported by money from the government.
It's common to talk about the "school system", which is the overall group of schools in a city, a state, or a country. For example:
The Franklin City school system is doing quite well these days.
So a "public school system" is a collection of schools that are all operated by the government. In the example at top, the speaker is talking about all schools in the country.