An exterior electromagnetic field generates a current in an ungrounded Faraday cage
to distribute electrons until the charged surfaces generate an opposing field
to cancel the field inside the cage. Outside the cage, however, the field
appears to travel through the cage.
A grounded Faraday cage still cancels an exterior field so that it has no effect inside the cage,
but doesn’t make the cage seem transparent to the field,
and it’s safe for someone on the opposite side of the field
to touch the surface of the cage.
See also in The book of science:
Readings on wikipedia: