There are some constants appearing in Maxwell's equations relating the electric and magnetic field. The equation with the constants is

Curl(B)=μ_{0}ε_{0}∂E/∂t, if you're mathematically inclined, where E and B are the electric and magnetic fields, respectively and μ_{0} and ε_{0} are two constants which had, prior to Maxwell, been discovered experimentally because of their relation to the electric and magnetic fields and charges and currents.

I take it you don't want to see the math involved, but what you do is manipulate this equation using a little calculus (and you have to invoke the other three equations as well, but those don't have the constants), to find that you get two equations which happen to describe the electric and magnetic fields as waves which move with speed c=(μ_{0}ε_{0})^{-1/2}. Maxwell correctly deduced that light was therefore a wave that had to move with speed c.