How does Edward Scissorhands qualify as an anti-hero?
You can consider Edward Scissorhands as an antihero in the sense that he is an untraditional hero. Some concepts of “antihero” invoke the antithesis of hero. I don’t think this applies because I don’t see how Edward is oppositional to heroism. But the idea of antihero is broad. Antihero can mean a character who is similar but peculiarly unlike a traditional hero. So, we could consider him the hero of this story. Aside from the daughter, I don’t know who else would even qualify as heroic.
Edward is passive, intensely shy and socially backward. Archetypal heroes have tended to be active, outgoing and socially commanding. But Edward does have nothing but good intentions and does his best to act on those intentions. He is at the mercy of a world that doesn’t understand him. So, acting on those intentions in an adversarial world makes him heroic. He is comparable to Frankenstein’s Monster, whom I would also consider an anti-hero.
The concept of anti-hero has really broadened over the years. Sometimes, even the antagonist can be considered an antihero. Two other examples that come to mind, for slightly different reasons, are Satan in Milton’s Paradise Lost and Othello.