i know only one method - to drag them manually, but when i do that i get the previous result as if i didnt cut them at all, seems like layers still hidden inside of this bars.
when you drag, click on the layer bar itself and not the edge of it. you will get an indication if you are on the edge by a bidirectional arrows cursor and you don't want that for this operation - you want the regular arrow cursor. hold and drag and press shift to snap and snap the layers edge to edge.
Editing layers and moving in and out points is one of the most basic AE skills you should have. It's not entirely intuitive but the technique is well covered in the help files and the community resources available directly from the Search Help field at the top right corner of AE. You should become very good friends with this tool and use it often, especially when something isn't working as you think it should. AE requires a bunch of study and some expertise to use efficiently. You'll never get there by just poking around in the UI or asking a few questions on the forums.
I know how to drag bars, I did it correctly, bit it didn't seem to work anyways. Maybe it was actually working, but I was in a hurry and didn't check properly. Anyways I did it all in Blender Video Editor.
I was asking different question if there is a way to remove all the gaps automatically, like in other software, Blender, Adobe Premier? Apparently no.
I know how to drag bars, I did it correctly, bit it didn't seem to work anyways.
if you drag bars, you are moving the layers and by looking at your timeline, you could be done with it in 5 seconds. by your description, you were dragging manually and getting the previous results. the only conclusion is that your were probably using the handles to move the layers. contributing to this assumption was the "hidden" layers inside layers description.
I was asking different question if there is a way to remove all the gaps automatically
you were asking for the best solution. manually would be the best solution in your case it because automatically isn't possible through the native tools in Ae. I think there's a script for it somewhere but can't be sure.
When you are dragging layers around holding down the shift key will snap the in or out points to the CTI, or to the nearest trimmed layer, or to the nearest keyframe as long as you have the four vertical bars showing when the layer is selected:
If you have a double ended arrow showing when you have a layer selected then you are moving the in and out point. The same thing applies, hold down the shift key while dragging and the in or out point will snap to the nearest, marker, keyframe, the CTI (Current time indicator) in, or out point. AE basics that everyone should know.
Moving clips around like this is also a lot easier to do in Premiere Pro. Looking at the original screenshot, you are moving clips around to tell a story. That is video editing. You do not want to edit in After Effects! Do your edit in Premiere Pro, then use AE's Import Premiere Pro Project command to bring your edit into After Effects to do any motion graphics or special effects stuff you want to do.