James Clerk Maxwell noticed that Ampere's Law was incomplete and needed a  term which allowed a changing electric field to create a magnetic field, even with no current or conductor present.  Maxwell already knew from Faraday's Law that a changing magnetic field could generate an electric field in empty space.  He then wrote down four equations that summarized everything known about the electric and magnetic fields and their relation to sources.  Then he realized that even if all source terms were set to zero (no charges, currents or magnetic materials) the equations still had solutions!