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.
It's possible that Voldemort has some grand plan that involves Harry. Rescuing him from Vernon is just the first step. He could be using this as a way to manipulate Harry's emotions and gain his trust. Once he has Harry's trust, he can then use Harry for whatever his ultimate goal is, like opening a powerful magical object or defeating a common enemy.
Voldemort could see Harry as a valuable asset. Maybe he believes that with Harry's power, they could rule the wizarding world together. By rescuing Harry from Vernon, he is showing Harry that he can offer protection and a different kind of 'family' compared to the Dursleys.
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.
Perhaps Voldemort has a change of heart. He realizes that Harry has some value to him alive other than just being his enemy.
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.
Maybe Voldemort uses his magic to stun Vernon first. Then he could simply apparate with Harry to a safe place.
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.
One possible motive could be his intense hatred for the wizarding world. Since Harry represents that world to him, he might murder someone he believes is part of the magic that he despises so much. Another motive could be fear. If he feels threatened by a magical being or situation, in his panicked state, he could resort to murder.
One possible motive could be fear. Voldemort is extremely powerful and if he threatened Hermione or her family, she might feel compelled to spy for him. Another motive could be a thirst for knowledge. Voldemort might have promised her access to dark magic knowledge that she couldn't get anywhere else.
One possible plot could be that Harry has to face his fear of Vernon in a very public and unexpected way. Maybe during a school event at Hogwarts, the boggart appears as Vernon and starts to humiliate Harry in front of his friends and classmates. This would force Harry to confront his past traumas related to living with the Dursleys and Vernon's cruel treatment of him.
Dudley might face his father's anger. Vernon could be furious that Dudley has gone against him to protect Harry.
It could be Sirius Black. Sirius is Harry's godfather and he is very protective of Harry. He might break out of his confinement (if he is still in that situation in the fan fiction) and rush to Vernon's house. Sirius has the skills and the determination to face Vernon and save Harry. He could transform into his Animagus form, a large black dog, and intimidate Vernon into letting Harry go.