Conceptual explanation of electrostatic shielding If you put a closed conducting surface in any external electric field, the field inside the conducting surface will be zero. 
A similar effect occurs if one has charged conducting surface without an external electric field. For that case you can argue on a conceptual level that the force on a test charge inside of the surface will be zero by using Coulomb's law (here a picture from Rogers: "Physics for the inquiering mind"):

You can sweep the whole spere with pairs of cones as indicated. Since the force acted by each particle on the test particle is proportional to the inverse square of the distance (Coulomb's law) and because the number of charges inside of the bottom of a cone is proportional to the distance squared, by superposition both contributions cancel out, so that the charges on area $A_1$ act on the test charge with the same but opposide force as that one from $A_2$. 
Now I am looking for a similar (in style and level) argument of how to explain my first statement, i.e. a non charged closed surface in an external electric field. Let the surface be a sphere (or general) and the field inhomogenic (using a square box and a homogenous field this is easy to).
Edit
As suggested in a comment, it the freely moving charges mus rearrange in such a way that the resulting field inside will be zero. I think it is conceptually clear that they rearrange in some way because the external field acts on them. However the point is, why do they reaarange in the "right way", i.e. such that the field inside is zero.
 A: When you place a closed conducting object in an external field the external field applies force on the electrons. Whatever be the shape of the conductor you will have a field inside in in the Transient State which lasts for a very short interval of time. This field applies force on the electrons. Since they are free to move , they move under the influence of the field and in turn produce their own field which lessens the total field inside the conductor. The process continues till the net field inside the conductor goes to 0. 
For a sphere in an external field , by this method of extending out a cone ,  the induced charges that come within the top surface of the cone , would be such that their current charge distribution gives total 0 field inside . Because That's the only possible solution and its the unique charge distribution. 
The same thing happens For a spherical shell not in an external field but having a net -ve charge which is at its center . The net field produced due to this charge in the shell would repel the electrons which will move outwards causing the net  electric field to 0 in the shell.
