You need to organize your work in AI better by using artboards and then exporting to a suitable format with the "use artboards" option enabled. The way it is now this will never work as you want it to. Your misunderstanding here is that AE has no way of knowing that your objects are supposed to be separate entities nor does AI provide a means of organizing it in such a fashion...
We don't know much about your illustrator file bit it is clear from your screenshot that you did not import as a comp and open the comp in the second screenshot. You have just added All your layers to a new comp.
It is also not clear what you want to do with your artwork. I don't know if the first screenshot is something you want in your final comp or if each rectangle is supposed to be a full frame.
Thank you for answers.
I did try to import file as a comp, but the effect was basicly the same. Unless I made it wrong. First printscreen shows storyboard for the animation to be, I definately don't want it to look like that. Each rectangle is supposed to be full frame. In each frame there are effects, vector graphics that are supposed to move.
So I make an artboard for first view and I export it, but i dont know is there any way and file type to have all elements (vectors) preserved when I import it to AE?
I kind of thought that was a storyboard. You need to make a separate illustrator file for each Frame of your storyboard. They need to be the size of the final comp. The only other option is to line up each element in its final position on a separate layer.
You do not have a screenshot that shows a properly prepared AI file imported as a comp. I am guessing you did not drag all your "released" layers above the original before trying import as a comp.
That was helpful, thank you