It can improve narration, but not in and of itself.
I am a big one for playing sim games with narrative elements and letting/encouraging people to describe their action, the environment, etc. (I credit the game Feng Shui with pulling me out of the D&D DM-control-freak ghetto on that).
Letting players know target numbers has minor gamist effects - for example, they know if they need to spend that hero point/mythic surge/whatever to make a roll, get an assist, etc. This gives them a little bit of an advantage, though frankly they and their characters should have a pretty good idea of that already if you're narrating well to them (e.g. "your strike hits him and digs into his armor but doesn't find flesh" vs "you whiff aimlessly in front of him" would indicate to me that a 1d6 surge might help the former...) But in the end - who cares? I certainly don't mind my characters doing (very) slightly better.
I will note narrating hits is hard without also knowing hit points. ("I slice off his arm and he bleeds out!" "No he doesn't, that 40 points of damage is just a scratch...) You will also want an explicit understanding that since D&D works off specific conditions, "I spin kick and knock them tumbling on their asses!" doesn't make someone prone or push them back unless the ability they're using does that - it's legit narration but your next sentence is "they leap to their feet and rush you," and no it doesn't take an action to stand up...
Too much worrying about "metagaming" here because you know hat you're rolling against is overblown. There's a lot of other games (and proficiency checks were like this in D&D 2e, and stat checks still are) where you're not rolling vs a DC you're just rolling vs your own stat or skill so always know whether you succeeded or failed per your roll by game design. Like GURPS and a huge number of other trad games, the "vs Difficulty" thing is less common really.
In fact, the main drawback of this stat exposure isn't just loss of the ability to fudge once stats are know, but for you to do fun add-ons. I know this will give the 3e/4e crowd the heebie jeebies but back in Basic/1e/2e if the monk spun kick the hell out of some guys you as the DM could declare they got knocked down or whatever to add interesting stuff to the combat beyond what the direct "application of hp damage by the rules" indicates.
You need to lead with narration yourself.
But just saying "He is AC 16 go" is contributing to the problem, not helping with it. Why would a player describe his hit with flair if that's the level of description you're putting in? It's way, way more effective to practice extensive description yourself, to where giving them the specific target number should usually be an exercise in validation ("Yeah, he sounded like he was encased in iron, AC 18 sounds fair.")
Numbers breed numbers, description breeds description.
Then encourage description explicitly. "Tell me how..." You can start with kill shots, it's pretty easy to get people into the Mortal Kombat type spirit by asking "OK, tell me how you take him out." "Oh I leap into the air and drive my spear through his rib cage and into the ground, leaving him impaled!"
Another trick I learned from FS was about environmental narration. Empowering the players to make up small stuff. If you're in a pizza parlor, sure you can pick up a pizza cutter and cut someone with it, you don't need to ask the GM if you can. In fact, I found that the usual RPG "say yes" advice wasn't as effective as "say no" - I told my players "If you ask me if there is a pizza cutter I'm going to say 'No, because you asked me instead of just doing it.'" That breaks them of their mother-may-I habit real quick.
Because in the end using description purely as a veneer over rules, especially a big rules system like D&D, has its issues; IMO you want to get to a place where the rules are secondary and are not a straitjacket. Players and GM should have some flexibility. The more you let the players have flexibility, you also get it. "Heck, you rolled a natural 20, if you want to knock 'em down or back or cut their face so they're not pretty no more, fine." And then when you want a badass villain to do something not "covered by the rules" you can too, without it being "unfair" or jarring.