It could be that Voldemort wants to use Harry for his own purposes against a greater enemy. Maybe there's a new threat emerging that Voldemort thinks Harry can help him deal with. So, he rescues Harry from Vernon to start using him in his own schemes. He might think that by having Harry on his side, he can gain more power or knowledge that Harry has access to.
Perhaps Voldemort has a change of heart. He realizes that Harry has some value to him alive other than just being his enemy.
In some fanfics, this could happen due to a different take on Voldemort's character. Maybe the author wants to explore a redemption arc for Voldemort. He might see something in Harry that makes him change his mind about killing him and instead decides to rescue him from Vernon's bad treatment. It could also be part of a complex plot where Voldemort has ulterior motives that initially seem like a rescue but are actually part of a bigger scheme.
Maybe Voldemort uses his magic to stun Vernon first. Then he could simply apparate with Harry to a safe place.
Well, in some fanfics, Voldemort might have his Death Eaters spying on the Dursleys. So they could've reported Harry's location to him. Another possibility is that Voldemort has some sort of dark magic tracking device on Harry that led him there.
One motive could be that Voldemort wants Harry on his side. He might think that by rescuing Harry from Vernon, who was clearly a bad caretaker, Harry would be more inclined to listen to him and join him.
One scenario could be that Voldemort has a change of heart. He might have had a vision or a revelation that made him realize that Harry is more valuable alive and not under the cruel treatment of Vernon. Another scenario is that Voldemort wants to manipulate Harry. By rescuing him from Vernon, he can start to build a relationship with Harry based on gratitude. This way, he can later persuade Harry to join his side or use Harry's power for his own purposes.
One reason could be that Voldemort wants to use Harry for his own gain. Maybe he thinks he can turn Harry to the dark side and use his power for some grand evil plan. He might believe that with Harry on his side, he can finally defeat all his enemies and rule the wizarding world completely.
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.
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.
Perhaps the civilians have been influenced by some anti - Naruto propaganda. They might not understand the true nature of Naruto and the fact that he has no intention of harming them. Also, they could be in a state of panic after some events that they wrongly blame on Naruto. For instance, if there was some minor destruction in the village that was actually caused by other factors but they assume it was Naruto's doing because of his connection to the Nine - Tailed Fox. This could lead them to try and harm him until Kakashi steps in to rescue him.
Dudley might face his father's anger. Vernon could be furious that Dudley has gone against him to protect Harry.