The public ( free ) education serves many purposes.
The most important one is to guarantee universal access to education for everybody, regardless of their material status. Earlier, education was limited to the wealthy only, and many so called poor people weren't able to pay the school fees.
Public education guarantees equal opportunities for all children, from all social, ethnic or religious backgrounds. No child is being left behind because it belongs to a particular group not welcome in a private school system.
Public education unifies a diverse population and prepares people for citizenship in a democratic society.