Leo left the house because he could not bear to be around Lily but not talk to her. Lily only found out about it when she woke up. She realized that it was probably Snow who said something. Lily was far to sensible for her age and she understood that her father had decided to take it upon himself.
How could she not know what Snow would have done to her. She was grounded when Leo was sick because she forcefully asked a classmate to share his candy. Snow was a noble and she could not tolerate such display from her daughter.
Don't get this wrong, Snow would have cut her heart out for her daughter, but she had to teach her the moral values. The little girl looked at the note and spotted a shallow water mark, she caressed it gently with her thumb and her gaze turned firm. She knew that Leo must have shed a tear while writing the note to her.