According to one source
Electric field is defined as the electric force per unit charge.
So another way to state your question is, why does a unit of charge (electron, proton, etc.) feel no force inside a Faraday cage?
In fact, it does feel a force from every unit of surface of the cage, according to the inverse-square law. Within a given solid angle, every piece of the surface within that angle exerts a force, but if the subject charge is closer to the surface, while the attraction per unit of area is larger, the area itself is smaller by the same factor, so distance from the surface makes no difference.
Then if you expand the solid angle to include the whole sphere, those forces all cancel out, so the charge feels no force.
The same argument explains why the shape of the cage makes no difference.
Gravity works the same way.
Put a mass inside a massive shell, and it feels no gravity, because it feels gravity in all directions, and those cancel out.