In fanfiction, Snape saving Harry often showcases a complex side of Snape's character. It might be a redemption arc for him, as in the original series, Snape had a very ambiguous relationship with Harry. In these fanfictions, it could be due to his love for Lily that he finally decides to fully protect Harry, which is a very interesting take on his character.
One possible aspect to analyze is Snape's complex character. In many 'Snape Saves Harry' fanfics, it delves deeper into his motives. Snape has always had a love - hate relationship with Harry. In these fanfics, his act of saving Harry might be due to his lingering love for Lily. It shows a different side of Snape, one that is more heroic and self - sacrificing than what we see in the original series at times. This also allows for exploring different plotlines and character development arcs for both Snape and Harry.
I also like 'When Snape Becomes Harry's Guardian'. In this fanfic, Snape saves Harry from a dangerous situation early on, and then takes on a sort of guardianship role. It has some really interesting interactions between the two characters as they learn to live with each other.
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.
One common theme is sacrifice. Snape may sacrifice his own position, reputation or even safety to save Harry. This is in line with his self - sacrificing nature for Lily in the original story.
One motivation could be Snape's sense of duty as a teacher. Despite his dislike for Harry in some ways, he still has an obligation to protect the students. So, when he sees Lockhart endangering Harry, he steps in.
Another possibility is that Snape uses his position as a teacher. He might threaten Lockhart with the consequences of harming a student. Given Snape's intimidating presence and his own power within the school, Lockhart would be afraid to cross him. Snape could also call for other teachers to help, making it clear that he won't let Lockhart get away with any harm to Harry.
In some versions, Snape might notice Harry's withdrawn behavior or signs of physical harm during Potions class.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.
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.
Well, he could use his magic skills secretly. Maybe he casts a spell on the Dursleys to make them docile or distracted long enough to get Harry out. For example, he might use a Confundus Charm on them so they don't notice Harry leaving.