A Faraday cage is a metal casing used to protect whatever is inside it from electric and magnetic forces. A well designed Faraday cage can protect a person from lightning as well as external radiation.
It is not a perfect insulator. Radio-waves and some magnetic fields can penetrate a Faraday cage with various degrees of attenuation. High energy photons such as x-rays are largely unaffected by a Faraday cage.
In the case of electricity, Faraday cages are easy to explain. The metal of the cage leads the electricity that strikes the cage to ground, without going through the interior of the cage. The cage is a sufficiently good conductor to take care of the electricity, and lead it safely to ground.
In the case of low energy radiation, the cage acts like a receiving antenna. It transforms the energy of radio-waves to electricity, which in turn is led to ground.
Such a cage will only protect against x-rays to the extent that it is built from a material sufficiently impregnable to high energy radiation.
The fact that low energy radiation is more efficiently shielded by a Faraday cage than high energy radiation can be explained in terms of momentum and pilot waves.