The translation above may not be the best, but you asked for an explanation, and that's what I am trying to give you.

The "tagiskola" is a fairly recent concept in Hungary, usually it refers to a small school (usually elementary) in a small town or village that belongs to a larger school in a nearby city. Think about it as a company with a branch office somewhere. The larger school has the administrative responsibility of the small school "branch", sometimes they share teachers (specialty teachers, such as art or music) may be commuting between the main school and the other campus.
This is an effort to make school districts more efficient, reducing costs, but still offer schooling for children in smaller, more isolated communities without forcing them to commute to the city.
Recently many small village schools lost their "independence" this way, they became "branches" or sub-campuses of other schools. Some argue for the benefits of this, some argue against the concept, it is a bit controversial these days.