Question

Why must electrostatic field at the surface of a charged conductor be perpendicular to every point on it?

Answer 1

Surface of the conductor is an equipotential surface for static distribution of charge on it. Had it not been so, then charge would flow between points with potential difference. But, then we can not have static charge on the surface.

Now, suppose electric field lines are not perpendicular to the local surface of the conductor . Then electric field will have some component parallel to the surface. This component will give potential difference $dV=-E\left(parallel\right).dl$ 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.