It was a little show of trust, Azula thought, the girl asking her to take her hand, asking that of a Firebender was dangerous if nothing else. Still she wouldn't harm the girl, there were far too many questions to be answered plus she had spoken of a place to stay. So the girl was safe from her.
And she was being all kind to the rude man. Azula didn't understand that but then people were sometimes very strange. She took the offered hand and noted that now that she had seen the man's burned arm the girl liked her even less.
"Indeed it does"
Still, she could firebend if the girl decided to try anything clever. "You have not told me your name" she asked conversationally, not at all used to standing in strange streets in stranger lands holding on to someones arm like a child would. "Or why you wish to help me?"