How does "off" fit in this sentence?
"She dashed off to her room" means that she went away from the place where she was and which isn't mentioned in the sentence, and dashed to her room. So with off there's the added meaning of going away from a particular place which isn't included in "She dashed to her room".
EDIT upon comment: There's the phrasal verb dash off which means write or draw something very quickly. Its use is totally different from the use of the preposition off described above.