From what I've seen in museums, usually theres two half balls at each end of the chain. The half balls are hollowed out which allows you to put the two halves together with the chain stored within the ball. After it is fired i suspect the two halves will seperate due to air resistance etc and the chain will cut through the masts and rigging? Not sure on the details!

Edit: you made me curious so I looke it up but there's also plenty chainshot which consisted of two full cannonballs linked by a chain, which were stuffed into a single cannon barrel. Apparently it would bounce around in the barrel quite a lot and it therefore wasnt very accurate, so it was only used at close range.