Since all of the grammar points have been explained by now, I'll add my take on the context you provided. Note that this analysis is pretty much overkill under any circumstances. There is no need to over analyze things like this so much, but it could help you out.
As Toritoribeさん wrote "だから means 勘違いじゃないって知っているから". This could help you "connect the dots", I think.
Notice that she didn't say 知っている
から, but 知っている
くせに. You may wonder about that change.
く せに (just like のに sometimes) implies that something after it was omitted. 知っているくせに. 知っているくせに just means lit. "despite the fact that you know...", but it implies criticism/contempt or disagreement about an action (listener's, in this case). You're free to imagine what ever makes sense after くせに, but the "omitted" part's implied meaning will be understood by everyone. No need ask about that. Here's where から comes in. から can be interpreted as "so" in this case, since that fits, but more often as "because". With that said, 知っているから can be interpreted as "you know (that) so,..." or "because you know (that),...". In both cases, the main clause (sentence) is missing. That missing part can be pretty much anything; "You know (that), so
doing something is wrong " or Because you know (that),
you shouldn't do something", etc.
Now, remember that implied "criticism" from earlier? Well, here's the tricky part: Since Harry knows she criticized his action, he attaches から, then replaces the whole phrase (知っているから) with だから and asks だからどうだって言うんだい. He's basically asking for the "missing sentence", i.e., 「知っているから、"missing sentence".」 He's doing that by using どうだ, which represents "contents of her intention" as Toritoribeさん suggested. That's basically all that's going on here. Again, as someone previously mentioned, all this can be kind of rhetorical and just signify Harry's frustration.
For the "English equivalent" of the phrase in question
, I would pick the one Mikeさん suggested - "So what's your point?". It feels to be the closest thing to what Japanese phrase conveys, while the suggestions from Majesticさん would work better from an English point of view (also in this context), in my opinion. I'm not going to preach to you about the best ways to go about studying these kinds of phrases since everyone has their own study methods, but I can tell you that trying to find every equivalent for every word, especially in set phrases, is not the best idea. The J→E will give you a lot of headaches, I speak from experience.
Note: I'm not an expert or anything, I'm still learning myself. Anyone is free to correct me if I'm wrong about this.