Perhaps Snape discovers some dark plot involving the Dursleys that could harm Harry more than he initially thought. The Dursleys were known to be cruel to Harry, and Snape might have found out that they were involved with some Death Eaters or dark magic users who planned to use Harry in a harmful way. So, he decides to rescue Harry to prevent this from happening.
At first, Harry might be extremely suspicious. Snape has always been cold and mean to him in the past. He would wonder what Snape's true motives are. Harry is used to being on his own and not relying on Snape for help, so this sudden rescue would be very unexpected.
It could make Harry more empathetic. Snape is a complex character, and through his rescue, Harry gets to see a different side of him. This could make Harry more understanding of others' hidden motives and pain. He might start to look beyond the surface when dealing with people, which would be a great development for his character as he faces more complex relationships in the wizarding world.
The plot twist could be related to Snape's true motives. He might be under an Unbreakable Vow to rescue Harry from the Dursleys. In the process, Harry realizes that Snape is not as bad as he seems. This could lead to a complete change in their relationship. Harry could start to trust Snape, and Snape, in turn, could teach Harry some unique magic that he has never learned before, which is crucial for the upcoming battle against Voldemort.
In the fanfic, Snape might use his knowledge of the Dursleys' house layout. He may enter the house unnoticed, find Harry, and then use a Portkey to quickly get them both out of there. He could also have some sort of charm or potion that he uses to counter any magic the Dursleys might have used to keep Harry there, like a spell - breaking potion. And he may have planned the whole rescue very carefully, choosing a time when the Dursleys are least likely to notice, like in the middle of the night.
Well, in some fanfiction scenarios, Snape might rescue Harry from Azkaban because he discovers new evidence that clears Harry's name. Maybe he uncovers a dark plot by some other powerful wizard who framed Harry, and his sense of justice, despite his past animosity towards Harry, drives him to act.
Love for Lily. Since Harry is Lily's son, Snape might feel an obligation to protect him. He couldn't save Lily, but he can try to keep her son safe.
Perhaps Snape uses magic to create a distraction at the Dursleys' place. He could make strange noises or apparitions that scare the Dursleys, and while they are in a panic, he grabs Harry and disapparates. This would be a quick and efficient way for him to rescue Harry.
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.
Maybe Snape was in a really bad mood and Harry happened to be in the wrong place at the wrong time. Snape is often portrayed as a complex character with a lot of pent - up anger, and Harry might have unknowingly triggered it.
After saving Harry, Snape may have to deal with a lot of mistrust from some people in Hogwarts who don't understand his motives. However, he might also start to form a closer bond with Harry, which could change the whole dynamic between them. This new relationship could lead to some unexpected alliances and enmities within the wizarding world. And if the Dursleys were to try and retaliate, Snape would have to constantly be on guard to protect Harry from any potential harm they could cause.