The more simple explaination is using the Gauss Law.When a conductor is charged , after equilibrium is reached , charges rest on the surface of the conductor , and inside a cavity , that is open space no charge resides , by Gauss Law , no electric field exists inside it and hence these cavity is kept protected from outside electrical influences , phenomena called "electrostatic shielding'.
The electric feild is determined by the way in which the voltage changes from
place to place. If you are inside a metal sphere, the voltage is the same
everywhere. Since the voltage isn't changing as you move around, the electric
field measures as zero everywhere inside the sphere.
As somebody else said, the electric field is only zero in an electrostatic situation, in which it is assumed that all the charges have had time to "settle down" before we look at the field. We also assume there are no currents flowing (obviously, a wire carrying current has an electric field inside, or else electrons wouldn't move along it).
If there was a field inside a conductor in electrostatic equilibrium, then the free charges in the conductor would accelerate in the direction of the field - which would mean the conductor wasn't in equilibrium - a contradiction. Hence, the conclusion follows.