The object dangling in the back of the F-35C is the tailhook, which snags hold of a cable on the carrier deck. The cable slows the aircraft down, allowing it to land efficiently and safely on the otherwise-dangerously short runways that aircraft carriers offer.
The technical requirements of taking off and landing from a carrier means that the F-35C is significantly heavier than the F-35A and F-35B variants. The C has an extra 208 square feet of wing to help create drag. Overall, the plane weighs over 6,000 pounds more than the other variants.
In November 2014, the F-35C conducted its first ever successful carrier landing. The landing came after nearly three years of delays due to tailhook design issues.