lexi and pyko give their snap answers to this storied question.
Close to zero importance, because I forget! And also I've found many games can work even without the narrative. For me narrative is more so a finishing touch or something to tie it together--I've yet to play a game where narrative is everything (or at least none is coming to mind).
In an escape room, narrative isn't a big deal for me, though a good narrative will edge a room up in my personal rankings. In a video game, it's pretty much everything for me. I've found that I get bored very quickly with games that don't have a strong narrative under the puzzles, and I likely won't finish them. That's probably the writer in me talking, though!