Why are we using off here? What additional meaning does it add?
The meaning of off from http://tfd.com/off that suits this sentence is: "away (from a place, time etc.)".
Learn English – Difference between “she cut her hair” and “she cut her hair off”
meaningmeaning-in-contextphrasal-verbsprepositions
Best Answer
There's more than a hair's breadth of difference between these two statements!
The first simply means she had a haircut. Maybe she is now sporting a new style, or maybe she just had a trim.
The second means that her hair has been cut drastically short, if not completely shaved. It might be used if she once had very long hair, but, after her latest trip to the salon, it's now cut very short.
Note that the difference between cut and cut off can vary, depending on what's being cut.
She cut her finger means that she's bleeding, and may need to put a bandage on it. She cut her finger off means that, unless a doctor performs reattachment surgery, she'll only have four fingers for the rest of her life. He cut five pages from the back of the book might suggest an editor opted for a shorter ending. He cut off five pages from the back of the book could mean a bookbinder removed physical pages with an exacto knife.
For hair, however, she cut her hair is simply a common expression meaning she got a haircut. She cut her hair off suggests something far more drastic, which I've already explained.