Fact: Vanya is adorable
Fact: Damian is a jerkface
Query: What is even going on with that?
OKAY SO. Some people may be scratching their heads going :? at Damian being nice and, should I even say it, liking liking Vanya. HERE'S THE LOW-DOWN.
Damian has a soft spot for children. He always has. I think he wants to be like them but doesn't know how, so he fulfills that by protecting them and relating to them. So much so that when he realizes he's in a room full of dead children posed in bed to make them seem alive, he totally hurls.
More than that, though, he does show self-sacrifice when it comes to another kid. Scarlet was kidnapped and mutilated and Damian couldn't save her in time, though she escaped. Later on, after she teamed up with Red Hood, tazed and kidnapped Damian, and generally was set up to be his rival, she was about to get shot. Damian jumped in front of the gun and got shot four times (hence the new spine). Laying paralyzed on the ground, he manages to say, "Tell the girl we're even." All of this after she threatened him repeatedly! Damian cares about kids, and he's willing to do stupid things to protect them.
So that brings us to Vanya. She's so pathetic to him. He hates that she needs constant affirmation and he hates that she doesn't take his word at face value. But he also needs to protect her. She's so sad and (by his perspective) she's totally being mistreated by her "family" and he hates that. She hasn't done anything wrong, so she shouldn't be treated poorly.
Another part of this dynamic is the fact that Damian is working very hard to maintain his reputation. In Gotham, Damian gets downtime from being... the way he is. Mostly because he has cases to work on and he can relax around Dick and Alfred. In camp, Damian is around people he doesn't trust 24/7 and that's hard for him. He's only ten and he has so few coping skills that he's started letting things leak. He tells stuff to Vanya because she's so pathetic and innocent that she'd never think of betraying him (he thinks). So he lets his guard down and instantly feels foolish and embarrassed which causes him to lash out. Vanya doesn't fight back, so he doesn't know what to do. He wants people to push him back, and when they don't (which is rare) he feels guilt. I don't really have canon for this, but I do have canon for him having vulnerable moments.
SPEAKING OF WHICH canon is deliciously inconsistent! Example: Last issue of Batman & Robin had Damian breaking up with his mother because he's so committed to the Robin mantle. That same week, Streets of Gotham came out, and he derided the position, saying that he didn't care about its reputation. HM. THAT'S FUNNY. So I kind of just go with what feels right. I tend to defer to the creator of Damian (Grant Morrison) but he is insane so it's a hodgepodge.
ANYWAY. I'M RAMBLING. The long and short of it is that it took a long time for Damian to open up to Vanya and he still sucks at it royally, but he also would protect her like there's no tomorrow, partially because she's pathetic, and partially because he's actually grown to like her. Maybe even like her like her.