Another motive could be related to his complex relationship with Dumbledore. Dumbledore might have asked Snape to look out for Draco, and Snape, being obedient to Dumbledore in many ways, would do so to fulfill his request. Also, Snape might believe that protecting Draco is part of a bigger plan to maintain the balance in the wizarding world.
Snape, being a skilled wizard, could use magic to protect Draco. He might cast protective spells around Draco when he's alone or in vulnerable situations. Maybe he uses a spell that alerts him whenever Draco is in danger, so he can rush to his aid immediately.
He could use his position of authority. Snape might confront the abuser directly. Since he's a teacher, he has the power to intimidate and put a stop to any immediate harm. He could also give Hermione a safe place to stay, like his own quarters (if it's within the rules of the fanfiction world).
One reason could be Snape's sense of justice. Despite his often - cold exterior, he might not tolerate Draco's bullying behavior towards Hermione.
Well, perhaps Snape uses his position as a teacher to step in. He could have been observing the situation secretly and when Draco was about to do something harmful to Hermione, Snape intervened with a firm command or some magic spell to stop Draco in his tracks.
One possible reason could be Snape's lingering resentment towards James Potter, Harry's father. Snape might see Harry as a reminder of the bullying he endured from James during their school days. So, he takes out his anger on Harry.
They might face magical justice. In some fanfictions, the Ministry of Magic could get involved and strip them of their magic, or at least put them under strict surveillance.
Well, Snape might heal Draco because of his own sense of morality. Despite his dark exterior, Snape has a code of ethics. Draco is still a student, and Snape may believe that every student should be given the chance to survive and learn from their mistakes. Additionally, Snape could be acting on behalf of Dumbledore in some way. Dumbledore often had complex plans, and Snape was his agent. Healing Draco could be part of a bigger plan to keep the balance in the wizarding world.
He could be doing it out of a sense of justice. Despite his family's views, Draco might see that the way the Dursleys treat Harry is wrong and decides to step in.
Snape would probably be very stern with Draco. He might give him a severe scolding and deduct points from Slytherin. Snape is a strict teacher and he won't tolerate such behavior from his own house students.
Snape could also be acting out of a sense of justice. Lucius was a Death Eater, and Snape, despite his own past as a Death Eater, had turned against Voldemort. Saving an innocent, especially Harry's daughter, from Lucius would be in line with his new allegiance to the side of good.