Question

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

Answer 1

As charge on the surface of a conductor are static (they do not experience any force), it is so only when there is no component of field along the surface of charged body.
Thus we conclude that electric field is normal to the surface.

Answer 2

-- If the field is not perpendicular to the surface of the charged
conductor, then there's some horizontal component of the field there. 

-- If there's any horizontal component of the field along the surface,
then the charges will move in response to the field, until the horizontal
component is zero.  They're free to move since we're talking about the
surface of a conductor.

-- So if there's ever a horizontal component of the field along the
surface, things quickly adjust themselves to make it zero, and the
field ends up perpendicular to the surface again.