Imagine for a moment that the electric field was not perpendicular to the surface. That means it has a component along the surface. Now, electric fields exert a force on charges, so now we have a force on the charges in the conductor along the surface of the conductor. This force isn't balanced by anything else, so it will then move the charge around. But that means that our system wasn't yet in equilibrium, since charge was moving around. In equilibrium, the charges must be at rest, and that can only be the case when there is no electric force along the surface, i.e., when it's perpendicular to it.