I think "stay with the feeling" means allow yourself to feel the feeling instead of trying to brush it away, stow it inside, or shrug it off. It means don't let yourself "change the feeling" if you get uncomfortable, similar to "changing the subject" if discussing something you want to avoid. Before I started therapy, I could not stay with the feeling. At the first hint of a feeling, I stowed it deep inside so I wouldn't feel it. Now I can stay with the feeling! The weird things we learn to do in therapy...

I don't know what "give the feeling some space" means. I haven't heard that one.