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.
Harry might be a bit conflicted. On one hand, he is grateful to be rescued from the Dursleys. On the other hand, he has a lot of emotional baggage regarding Snape. He might struggle between his natural inclination to be polite and grateful to his rescuer and his past experiences with Snape that make him hesitant to fully accept Snape's help. It would be a journey for Harry to come to terms with Snape's new role in his life.
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.
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.
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.
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.
Well, he might be quite reluctant at first. Snape has always had a rather complex relationship with Harry. He might see it as a huge burden and be full of doubts. After all, he has his own grudges and past issues with James Potter, Harry's father. But on the other hand, he also has a sense of duty towards Harry as he was Lily's son. So it's a mix of emotions that he has to deal with.
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.
Snape could initially be skeptical. Given their long - standing enmity, he might not believe that Harry's apology is sincere at first. He might respond with a cold glare or a sarcastic comment, like 'Potter, your apologies are as empty as your head.'
Snape might initially be surprised. He could then use this knowledge in a rather Snape - like way, perhaps to gain an advantage in his complex relationship with Harry. Maybe he'd be a bit more guarded in his interactions with Harry, or he could even try to subtly help Harry overcome his fear while still maintaining his tough exterior.
In some fanfictions, Snape might rescue Harry out of a sense of duty. Snape has always had a complex relationship with Harry. He could rescue Harry from a Dark Magic attack. Maybe he realizes that Harry is crucial in the fight against Voldemort and despite their personal differences, he steps in to save him. This could also be a turning point in their relationship, leading to a more understanding and less hostile dynamic between them.
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.