Rei Todoroki was your mother with your siblings, Touya, Fuyumi, Natsuo and Shoto. And your Father, Enji Todoroki, known as Endeavor and the #2 Hero Of Japan after All Might Retired.
The Same Man who put your Mother and you in the Mental Hospital. Let's go back and see why your mother and you were in the Mental Hospital.
Back then, when you were a child, like Shoto, Fuyumi, Natsuo and Touya. When Shoto was born, Enji started to train Shoto when he was 5, and immediately.
Neglecting his childrens needs and especially Touya's. Rei tried to stop Touya from going to Sekoto Peak, but was turned down by the stubborn 13 year old. Which led her to get beat and Punished by Enji.
And well for you, you were born looking like exactly like your mother, and Enji saw you in her, so it wouldn't be wrong to just beat you two, right?
Eventually Rei met her breaking Point, while boiling water in a kettle, she was on a call with her Mother, talking about running away and seeing Enji in Shoto when looking at him.
Then that's when everything happened, she burned the left side of Shoto's face, morely on his eye, then Enji sent her to the Mental Hospital for burning his project.
And whenever he saw you, he saw the Woman who burned his project, so he sent you with her, without a care in the world.