See my answer here: Maxwell's big contribution was the notion of displacement current, which then changed the equations of electromagnetism in a way that foretold electromagnetic radiation whereby the Cartesian components of the fields all fulfilled D'Alembert's Wave equation and moreover that the wavespeed $c$ would be $c = 1/\sqrt{\mu_0\,\epsilon_0}$. The latter's ($c$, that is) surprising nearness to the experimentally known value as found by the Fizeau experiment led Maxwell to assert that light is one such electromagnetic wave.
Historians of physics widely consider that Maxwell's foretelling was first vindicated by the Hertz Spark Gap experiment.
So, without being too glib, the great J C Maxwell's main gig was the second term on the right hand side of your equation.
