That's a bit (a lot) over my head.
You bike, right? Imagine the chain of your bike, with a gear at each end. Well, in this case instead of powering a wheel, it powers the screw. And instead of going forward parallel to the ground, it goes up, and receives power from a handle rather than peddle (the only difference being that it's powered by hand, rather than ped, or foot). That's the basic mechanism he's discussing.
And then instead of receiving the power from the handle directly, it receives it from a gear which is powered by a perpendicular gear to which the handle is attached, so that the path of the handle looks circular from above, to dodge the waggling appearance of the handle in the previous animation.
He's saying pop that mechanism on a thingamajigger like what Manveru made, and you'll have something that would work in real life, is consistent with its function in the game, and visually you can tell more or less what's going on.
It a screw, a tube section and a block. That's it, according to DF. Even the handle I invented, because it's on pictures of old RL archimedes screws. Certainly no chain or belt involved.
Like forging tools with no hammer or tongs, lot of things have to be thought of as abstracted. A chain might be going a bit too far since you'd think that would be hard to manufacture at dwarven tech level, but the items needed to construct something can't be much more than a guideline. Still, with regards to the chain specifically, you could also just have two connected gears, one connected to the screw (presumably at the end, by means of a short axle) and another above, interlocking directly without the aid of a chain.