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.
It could also be that Snape has some sort of jealousy. Harry is the Boy Who Lived, the center of attention at Hogwarts. Snape, who has always been a bit of an outcast and has his own dark past, might be envious of the adoration Harry receives, and this leads to his abusive behavior.
In this fanfiction, Harry might stand up to Snape's abuse by using his wits. He could expose Snape's unfair treatment in front of the other professors. For example, he might gather evidence of Snape's partiality in grading or his excessive use of punishments towards Gryffindors. By presenting this evidence to Dumbledore or other authority figures, Harry would be taking a stand against Snape's abuse.
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.
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.
One possible reason could be that Harry is afraid of Snape's intimidating presence. Snape has always been strict and rather unfriendly towards Harry in the original series, so in the fanfiction, this fear might be amplified, causing Harry to run. Another reason could be that Harry has stumbled upon something he shouldn't have seen that involves Snape, and he's running to avoid getting caught.
Perhaps in this fanfiction, Snape shows a side of himself that is completely different from what Harry has known. He might be more open and honest with Harry, sharing secrets that make Harry believe he can be trusted. Like Snape might disclose some long - held information about Harry's parents that only he knew, which makes Harry think Snape has been on his side all along.
In some versions, Snape might notice Harry's withdrawn behavior or signs of physical harm during Potions class. Maybe Harry flinches when someone gets too close, which makes Snape suspect something is wrong.
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 possible reason could be copyright infringement. If the fanfiction uses too much of the original 'Harry Potter' material without proper authorization, J.K. Rowling (or those in charge of the franchise) might take legal action. Another reason could be if the fanfiction misrepresents the characters in a way that goes against the intended spirit of the original work.
It could be that in the context of the fanfiction, Snape believes Harry has been disrespectful towards him or other teachers. For example, if Harry were to talk back or show disobedience in class, Snape would likely punish him as a way to assert his authority.
One reason could be that he realizes Harry has a lot of potential and wants to nurture it. He sees that Harry is in a difficult position, being the target of Voldemort, and decides to help him despite their past differences.