Todoroki's scar is a distinguishing part of his character, marking his face like a horror movie character. But contrary to what most people might assume, this outcome was not achieved in battle. Due to an accident driven by the trauma that resulted from his mother's mental oppression by Todoroki's father, Endeavor (who runs the Hero Institute), Todoroki Houka ended up with ugly scars all across his body.
Because she had been emotionally abused by Todoroki's father Endeavor, and was at her wits' end from an existence weighed down with anxiety and fear, she mistakenly scarred his face with boiling water. Because when she saw him, she was horrified to think that he was just like his dad.
As he tries to sobriety get up from the darkness and pain of his past, Todoroki is just Shantering meaning of both his existences: one domesticated existence (the one which he converted into light), and another that comes straight from the ground and through up enough power to gag his spider-strings life as an Author. He has no way to escape the responsibility grown up from such a life of abuse. A constant reminder of cruel memories and injury is thus Todoroki's facial scar which he received when his mother was tmost afraid.