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.
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.
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.
Perhaps in this fanfic, Severus doesn't just save Harry physically from the Dursleys. He also saves Harry from the emotional and psychological damage that the Dursleys have been inflicting on him. Severus, having his own traumatic past, might be able to relate to Harry on a deeper level and starts to mentor him in ways that Dumbledore never could. This could lead to a completely different Harry, more confident and less impulsive.
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.
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.
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.
Perhaps Severus, while rescuing Harry, stumbles upon a hidden connection between the Dursleys and the Death Eaters. This would add a whole new layer of danger and mystery. He might have to fight off some of his former allies to get Harry to safety. It could also lead to a change in Severus' perception of Harry as he realizes the boy has been in more danger all along than just from Voldemort.
One possible plot could be that Voldemort realizes Harry's importance in a different way. Maybe he sees that Harry has some power or knowledge that could be useful to him in the long run. So, instead of wanting to kill Harry, he decides to rescue him from the Dursleys to gain his trust and manipulate him later. For example, Voldemort might think that Harry has a connection to some ancient magic that only he can access, and by saving him, he hopes to get access to it through Harry.
Well, perhaps Fleur uses her Veela charm to intimidate the Dursleys into letting Harry go. She could arrive at Privet Drive in a blaze of magical beauty and power, and the Dursleys, being the cowards they are, would be too scared to oppose her when she demands Harry come with her.
Well, perhaps Daphne somehow discovers Harry's awful situation at the Dursleys. She might use magic to break into the house secretly. Then she could use a spell to make the Dursleys forget about Harry so that she can take him away without a fuss.