A Faraday cage is best understood as an approximation to an ideal hollow conductor. Externally or internally applied electromagnetic fields produce forces on the charge carriers (usually electrons) within the conductor, generating electric currents that rearranges the charges. Once the charges have rearranged so as to cancel the applied field inside, the currents stops.
If a charge is placed inside an ungrounded Faraday cage, the internal face of the cage will be charged (in the same manner described for an external charge) to prevent the existence of a field inside the body of the cage. However, this charging of the inner face would re-distribute the charges in the body of the cage. This charges the outer face of the cage with a charge equal in sign and magnitude to the one placed inside the cage. Since the internal charge and the inner face cancel each other out, the spread of charges on the outer face is not affected by the position of the internal charge inside the cage. So for all intents and purposes, the cage will generate the same DC electric field that it would generate if it was simply affected by the charge placed inside. The same is not true for electromagnetic waves.
If the cage is grounded, the excess charges will go to the ground instead of the outer face, so the inner face and the inner charge will cancel each other out and the rest of the cage will retain a neutral charge.
Effectiveness of shielding of a static electric field depends upon the geometry of the conductive material. In the case of a nonlinear varying electric field, and hence an accompanying varying magnetic field, the faster the variations are (i.e., the higher the frequencies), the better the material resists penetration, but on the other hand, the better it passes through a mesh of given size. In this case the shielding also depends on the electrical conductivity of the conductive materials used in the cages, as well as their thicknesses.