In many fanfictions, Snape saving Hermione from Voldemort can be a very interesting plot. It could be due to Snape's complex character. He was a double - agent in the original series. So, in fanfiction, his loyalty to Dumbledore might drive him to protect Hermione, who is an important part of the good side against Voldemort.
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).
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.
Well, it could be that Snape has been secretly observing Ron's erratic behavior towards Hermione. When the situation gets out of hand, he intervenes. He might use his position as a teacher to discipline Ron and in the process, saves Hermione. Snape is a complex character with a lot of power and knowledge, so he could use various means like his occlumency skills to sense the danger Hermione is in and act accordingly.
One reason could be Snape's sense of justice. Despite his often - cold exterior, he may not tolerate Ron's unjust behavior towards Hermione. Another reason might be that he sees something in Hermione that he values, like her intelligence or her thirst for knowledge, and doesn't want it to be extinguished by Ron's actions.
One possible reason could be that Snape sees Hermione's value as a brilliant student. He might recognize her potential and not want it to be wasted just because of her dad's actions or situation. Maybe he has a sense of justice deep down, despite his often cold exterior.
In many such fanfictions, Snape might notice Hermione's change in behavior. For example, she could become withdrawn or less enthusiastic in class. He's a very observant character, and her academic performance might also start to decline. He could pick up on the signs during one of his patrols when he sees her looking distressed or if he overhears something that makes him suspect she's in trouble.
For Harry, saving Snape could make him more empathetic towards Snape's past actions. He might start to understand the complex web of loyalties that Snape was caught in. It could also make Harry more confident in his ability to make moral decisions independent of others' opinions.
In some fanfictions, Draco might have a change of heart due to his hidden feelings for Hermione. He could use his knowledge of the Death Eaters' plans and Voldemort's weaknesses to rescue her. Maybe he has some secret magic or an unexpected alliance that helps him get to Hermione just in time.