Odd spot for the trigger to me. I would think you want the target under the falling part when triggered not off to the side. I must be missing something big time.

Can somebody explain, how the part ot the trigger-system highlighted in red works?

Watch alone. I watched an episode on Netflix. If you really live off the land you need to kill big animals. Small animals are a fine supplement but large animals are needed for the long haul. fortunately KS is chock full of cattle.

Can somebody explain, how the part ot the trigger-system highlighted in red works?

That is a type of toggle trigger.
f is the toggle and it is stuck in the ground just enough to hold the log up.
When strings are pushed the bottom part of f (where the strings wrapped around the stake are attached to f)is pulled enough to fire the trap.

you can build a kulemka-type deadfall with a chelak-type trigger,

The target animal is under the log when they hit the trip line (g) in the diagram. g extends to the trigger that is off to the side and dislodges it and the log falls. The target animal does not have to make contact with the trigger so location of the trigger under the deadfall log is not necessary. The trip line activates the trigger and releases the deadfall.

Can somebody explain, how the part ot the trigger-system highlighted in red works?

It's also better not to attach any of the parts to standing trees, or lean the weight log against a standing tree, because it will set off the deadfall in windy weather. Some people praise this kind of deadfalls but it's mostly overly enthusiastic amateurs with youtube channels %) In reality, they're tedious to build, cumbersome, and very vulnerable to being set off by jays and heavy snow, especially now that winters have been becoming milder throughout most of Siberia, meaning fewer persistant high-pressure systems and more frequent snowfalls. They have an advantage over legholds in that they're still legal here (all legholds have been banned by animal rights nazis about 15 years ago, disguised as a move towards Euro-integration). Another big disadvantage is that you can't move them, which is a big problem in Eastern Siberia/Yakutia with frequent wildfires, meaning if you rely on deadfalls, you may have to rebuild them every few years, so for people with very big traplines it's not an option. It only makes sense to make a few in spots wtih really high marten/sable traffic, such as dispersal bottlenecks or ravines along rivers.
There's an alternative trigger mechanism, with a "pedal" stick, which is better because you can make it on the spot, but requires more time to tune to be at least a bit resistant to birds. I used to have a few on my trapline but it was mostly to try building/using them.