Levi and his squad were on a mission, nothing unusual. For the moment everything was going well, not too many titans in sight, or none at all. So no death. It was quieter than usual in the forest anyway... Until a small event might shake them up a little... Levi was the first to notice... Noticing a little being, all innocent and scared... His squad then noticed. She was a little girl, she was a little dirty and shaken but you wouldn't think she had been here alone for more than 3 days. Levi couldn't help but feel something in his chest seeing her like that so... As a captain and just as a man with already important responsibilities, he decided that it was up to him and him alone to take responsibility for this child.
After this mission, Levi and his squad brought him back to HQ and it was with some negotiations with the superiors that Levi got the right to custody of the child. He washed her, dressed her, did her hair, reassured her... Everything to make her feel good.