How can an aircraft not fall into the sea after the arresting cable snaps during a carrier landing?
The truth of the matter:  That E-2C pilot escaped crashing by a hair's breadth.  He was just very, very fortunate that the combination of holding full power, the position on the flight deck where the arrester pendant failed allowing for a small takeoff roll to accelerate, plus the 40+ ft drop off the deck of the ship to the waves combined with the ship steaming into the wind was enough to gain him the end airspeed he needed to fly again.  The flight crew received the Air Medal for their heroism in the face of that situation.  As an E-2 does not have crew stations equipped with ejection seats, there was no other option available to the flight crew than to attempt to gain airspeed and hope for the best.  That cross deck pendant failure could have easily resulted in disaster and most pilots in that situation don't live to tell the tale.

Although it is rare, cross deck pendant failures can and do happen and are extremely dangerous, both to the aircrew of the recovering aircraft and to nearby flight deck personnel; there have been cases where a failed cable has literally sliced a man in half.  As a precaution, each cross deck pendant has as pre-set service life of 100 traps, then it is removed from the purchase cables on the arresting gear and destroyed.

This video was shot by an sailor aboard the USS Ronald Reagan during sea trials in 2003.  The arresting gear failed, sending the aircraft over the edge of the deck.  The F-18 pilot ejected and survived.  Note the amount of damage which the end of the cross deck pendant does to equipment and personnel as it whips across the flight deck.