Interesting video with some good details (despite the time wasted on fluff conversation).
Two concentric clutches driving to coaxial input shafts, on inside the other which is hollow. So that's how they do it, a neat solution, probably the only one that makes sense.
Gear selection seems pretty conventional for a manual gearbox.
The really interesting feature is the use of two secondary shafts, each split into two coaxial sections, one section carrying odd numbered gears and the other carrying even numbered gears. The split of each is what allows the dual-clutch feature to pre-select a gear. A traditional 4-speed box would have only one secondary shaft, which of course is not split. It would carry all the gears, possibly using direct drive for 4th.
Using two secondary shafts would seem to be a way to squeeze more gears into a smaller space, and would accomplish the same thing even if it wasn't a dual-clutch transmission. They mentioned it only briefly in the video but both shafts seem to be directly connected to the equivalent of the differential ring gear. One shaft holds the lower gears, the other holds the higher, but would actually use different size drive gears to create different ratios for the two shafts.
They did not describe what sort of fluid makes it a wet clutch. The alternating drive and driven plates in the clutch are like the old New Departure coaster brake on bicycles, effective and robust.
Finally and predictably, gear selection management is handled by a computer control unit that anticipates the gear based on the driver's behavior. All straightforward. It uses electromagnetic pickups to keep track of what gear is selected. That's not necessarily the only way to do it but seems pretty simple and effective.
Takeaway number one - gearboxes are non-trivial!