I don't have an exact answer, but I read the theory somewhere that a hero gets staggered after receiving X% of their max health as damage (check posts around where Sven previewed Cutie's legendary). I've been watching for that ever since, and it seems to hold true. Someone who reads the game files might be able to find something defining the stagger damage threshold. I think it's around 30-40% from a single source, but there might be more to it. Don't take my word for it.
Edit: And what is stagger? I think of it as an animation cancel or override. Whatever the hero was doing before they were staggered gets overridden with a recoiling animation. A guildmate called it "mini-stun". I've seen Orc Monk get staggered to cancel his heal where there's a faded green bubble on-screen that didn't burst as it normally would. It's like he was 60% through that skill's animation when he got staggered, but his target only gets healed at 70% so no one gets healed and he moves on to the next attack/skill in his attack sequence. Stagger is also one cause for Medusa not stoning the entire enemy team, and I just don't like that behavior.