платный - reqiure payment, it's adjective
я учусь на платном - я учусь по платной форме обучения
There remain practice, when government pay for a student from budget for his high grades. At least, it must work such.
Also, there is form of getting education when student pay for himself, it's платная форма обучения, i.e. он учится на платном (на платной)