It could be Voldemort. He might have influenced the Dursleys in some way to harm Harry, and Severus has to save Harry from Voldemort's long - reaching plans. For example, Voldemort could have placed a curse on the Dursleys' house that would eventually harm Harry, and Severus discovers it just in time.
Obviously, Severus Snape and Harry Potter are the main characters. Severus is the one doing the saving, and Harry is the one being saved from the Dursleys. But there could also be other characters involved. For example, Dumbledore might make an appearance. He could either support Severus's actions or be a bit wary of them. Petunia Dursley could also be a significant character as she is part of the reason Harry needs to be saved from that household.
Maybe Severus discovers some dark magic plot that involves the Dursleys harming Harry in a more serious way. So, he intervenes just in time. He could use his skills in potions and magic to subdue the Dursleys if they resist, and then take Harry to a safe place, like his own secret hideout or a friendly wizard's home.
In some fan fictions, Severus might save Harry from the Dursleys because he realizes that Harry is the key to defeating Voldemort in the long run. He might see that the Dursleys' treatment is making Harry weak and less likely to be able to face Voldemort. So, he intervenes, maybe by intimidating the Dursleys into treating Harry better and providing him with proper education about the wizarding world. He could also give Harry some potions to boost his health and magical abilities that have been stunted by the Dursleys' neglect.
In some fan fictions, Bill might use his experience and magical skills. He could arrive at the Dursleys' place under the guise of some official wizarding business. Once there, he realizes the horrible situation Harry is in. Maybe he convinces the Dursleys that it's in their best interest to let Harry go with him, using a bit of magic to persuade them subtly.
There's also the idea of envy. Voldemort might envy the normal life that Harry could have had if not for the wizarding world's interference. By saving Harry from the Dursleys, he is both taking Harry away from a life he could never have himself and also trying to create a new situation where he can be in control of Harry's life, rather than the Dursleys or the Order of the Phoenix.
Well, perhaps in the fan fiction, Severus saves Harry from the Dursleys but then has to deal with the consequences of going against Dumbledore's plan. Dumbledore might have had a reason for leaving Harry with the Dursleys, and Severus' actions could set off a chain of events that changes the whole wizarding world. Also, Severus could find out that the Dursleys were part of a much larger conspiracy against the wizarding world, and by saving Harry, he uncovers this plot.
In the fan fiction, Severus could start to develop a sense of responsibility towards Harry. At first, he may have saved Harry out of a sense of duty to Lily Potter, but as time goes on, he realizes that Harry has his own worth. This could make him more open to forming a bond with Harry, which is a huge change for his character who has been so closed - off for so long. He may also start to question some of his long - held beliefs about the Gryffindors, especially as he sees Harry's courage in the face of his difficult upbringing at the Dursleys.
The Dursleys might be left in a state of shock and confusion. Since they have been so mean to Harry, they could be worried that the wizarding world will come after them for revenge. But since they are Muggles, the wizarding world may just leave them be, but they will always live in fear of the unknown.
In this fanfiction scenario, Bellatrix saving Harry from the Dursleys is quite an unexpected twist. It could be that Bellatrix has some ulterior motive. Maybe she wants to use Harry for her own dark purposes against Voldemort. She might see Harry as a valuable pawn in the power struggle within the Death Eater ranks.