Her brother? Ah, then that's good, if she's with family, and it sounds like it may be an older sibling as well. Izuku relaxes, then nods his head with a smile as she explains further. What a smart girl...she speaks almost like an adult in some cases, and it sounds like she had a rather busy lifestyle before. Communal living...maybe she also stayed in a dorm? Like how he does at UA? Or it might be something entirely different.

"I see! I'm glad your brother is here to keep you company-ah! R-Right, I'm sorry! I'm Izuku! Midoriya Izuku."

He bows forward, a small blush on his face as well. Forgot to even introduce himself, how embarrassing!