Surface of the conductor is an equipotential surface for static distribution of charge on it. If it is not so then charge would flow between points with potential difference inside the conductor. But, then we can not have static charge on the surface of conductor.
Now, if electric field lines are not perpendicular to the surface of the conductor, then electric field will have some component parallel to the surface.
This component will give potential difference dV=-E( parallel).dl between the nearby points. Then, the surface is not equipotential surface. So, for surface to be equipotential surface for static charge distribution the field lines at the surface are perpendicular to the surface.