Character Analysis
We've seen Bellatrix Lestrange before: She's the witch who killed Sirius. She's also Narcissa Malfoy's sister/Draco Malfoy's aunt.
Bella is very loyal to Voldemort and eager to defend "her" Dark Lord against all enemies. That's why she's so suspicious of Snape at the beginning of the film: even though the Dark Lord has decided to trust Snape, Bella wants to make sure he isn't going to double-cross them with Dumbledore (who believes Snape is loyal to Team Good Guys).
So, when Snape agrees to help Draco Malfoy with his mission, Bellatrix makes Snape take the Unbreakable Vow (which is basically just a magical way of enforcing the promise).
Now, is it her business to butt into Narcissa's chat with Snape? Probably not—but Bella is generally pretty aggressive about taking on the Dark Lord's wants/needs as though they're hers. Also, tbh, she seems like a straight-up weirdo.
We should also mention: Bellatrix and Harry have a run-in when she and other Death Eaters show up at the Weasley house. Despite the fact that he's generally a peace-loving kid, Harry is eager to hurt Bellatrix after what she did to Sirius, and he and Ginny end up hunting her.
To no avail, though. While they're out in the fields searching, she and the other Death Eaters go back and destroy the Weasley home. Did we mention she's a nasty piece of work?
Well, she is.