I read an article on this where the showrunners actually explained the reasoning behind it. It's actually a call back to Season One. Not that I can remember but Ned Stark said to Arya that one day she will be a lady and get married to which she replied "No. That's not me"
When she see's her direwolf, she obviously wants it to come back home with her and be a loyal companion. But the direwolf, recognising her, doesn't want that as "That's not her", it's clearly moved on doing it's own thing and leader of a pack.
So by Arya saying "That's not you" to her direwolf, even though she's kinda heartbroken, she realises that the wolf is only doing what she would do if she was the wolf.