As other people have said, this isn't a particularity of 4e especially. Even in 1e and OD&D/Basic hit points were pretty abstract. Your average old D&D 3rd level fighter could eat 3 longbow arrows before he went down. So obviously even in that game hit points were entirely abstract.
So, yeah, we just use whatever 'fluff' is appropriate at the time. Most HP loss is just fatigue, loss of morale, using up luck, using up magical protection, etc. When you hit bloodied you probably want to describe your character as being wounded in some way.
Healing could be many things. A potion or a prayer magically knits flesh and restores vigor. An Inspiring Word increases your morale and resolve to fight. The cleric infuses you with some divine magic, the warlord just yells at you to "keep fighting, you worthless grunt!" or something. AD&D definitely tended to fluff ALL healing in magical terms, but the concept was similar.
Healing after a fight or if someone stops to use Heal on you? Could be talk, a sip of water, a slap on the face, a little minor magic, or a quick sticky bandage.
Generally think of characters being like Bruce Willis in Die Hard. You get cuts, bruises, burns, whatever, but at the end of the scene someone slaps a bandage on it, you limp a little for a few minutes, everything is fine. Later when you hit home you can wince, remove your bandages, get the maid to pour you a hot bath, and have the cute innkeeper's daughter sew up your cuts.
Just IN GENERAL, 4e is very loose about HOW things happen, deliberately. Think about skills. Is the wizard picking a lock? Yeah, little minor magic there. Fighter intimidates someone? Well, those iron muscles put a fright in them. You can always fluff stuff as 'magical power', 'divine fortune', whatever you want. That deals with a lot of "how could I do that?" kinds of questions. It is a magic world, stuff happens. Heroes have all kinds of little tricks up their sleeves, minor magic, whatever they need. That's why they're the stars of the show. If they weren't fated to do crazy stuff they'd be minions somewhere and you'd be playing some other character.
You can also retcon things a bit. Come and Get It, well that orc fell prone AFTER he charged you for insulting his mother (and you beat him down). Forced movement often fluffs that way. The bard isn't necessarily shoving everyone around the grid, he's just baiting the enemy into making bad moves and directing the PCs to make better ones. So it can often be "Nope, that orc isn't over there, he went OVER HERE!" Remember too, turns are a crude approximation of the sequence of events.
Finally, the rules outright tell the DM to do what makes sense. If a situation seems nonsensical you can just change what happens. I wouldn't do it much and only with good reason, but as a last resort it certainly works. The rules can't possibly cover everything.
Anyway, none of this is really entirely unique to 4e. All of the same narrative devices have been used since the days of yore (1974 anyway). I guess more could be said about this kind of thing in the rules, but no other edition of D&D ever seems to have addressed it much either. Gygax mentions that hit points are abstract once in the 1e DMG IIRC. That's about it.