You've seen the drawings of how the anti-rollback device works, right? Basically a hinged lever that falls down into the sawtooth on the lift, engaging with each tooth as the train goes up the hill.
Well, the chain clutch works in exactly the same way. The chain clutch is a hinged lever with a hook on the end that catches on the link of the chain. I refer to it as a "chain clutch" because it is, essentially, an over-running clutch mechanism. If the train is moving faster than the chain, as at the base of The Beast's second lift, for instance, the lower edge of the pawl will push against the upper edge of the chain link, pushing the pawl out of the way and allowing it to pass to the next link. So the train can freely outrun the lift chain, which becomes particularly important when it goes over the crest of the lift. But if the chain is moving more rapidly than the train, the lower edge of the chain link will catch against the upper edge of the lift pawl, pushing it...and the train...on up the hill.
More detail than you probably need, but there you have it!
--Dave Althoff, Jr.