I am having a hard time to try and explain this but the jist of it is that the whole chop and parallel bar move together simultaneously. The chain ensures that both move at a steady rate. When installed properly of course. Some people comment that it does not move as fluidly as a normal parallel bar would (some resistance when turning the wheel).
Tightening the wheel moves the parallel bar back into the bench, as it is attached to the chop, which pulls the chain. The chain can never slack as the distance rung never changes. That I think is the important part. After reading information on several sites I could not find straight information about the movement of the chain.
In the build that you got the picture from you will see that some roller wheels were installed to help guide the bar and prevent racking.
Proper attribution is important. I found this image from a lumberjocks post of a gentleman in Russia outfitting his bench with this vise apparatus. It actually is a kit that is sourced from the US of mostly US parts.
You can find the kit and all instructions for its installation at Anchora Yacht Service woodworking site. You can also find, on that page, a link to another installation showing.