It could be that Voldemort has a change of heart due to some past connection. Maybe he realizes that Harry is related to someone he once cared about in a way. So, he decides to save Harry from the Dursleys. He may have found out that the Dursleys were planning something truly evil for Harry, like sacrificing him to another dark entity. Voldemort, seeing this as an unacceptable act even for him, swoops in and rescues Harry.
Perhaps Voldemort discovers that there is a greater threat than Harry that is somehow related to the Dursleys. So, in order to protect his own interests and the wizarding world (in his own twisted way), he saves Harry. He might see that the Dursleys are involved with a dark force that could destroy both the Muggle and wizarding worlds, and Harry is an important piece in stopping it. So, he takes Harry away from that dangerous situation.
One possible plot could be that Voldemort realizes Harry's importance in a different way. Maybe he sees that Harry has some power or knowledge that could be useful to him in the long run. So, instead of wanting to kill Harry, he decides to rescue him from the Dursleys to gain his trust and manipulate him later. For example, Voldemort might think that Harry has a connection to some ancient magic that only he can access, and by saving him, he hopes to get access to it through Harry.
Well, perhaps Fleur uses her Veela charm to intimidate the Dursleys into letting Harry go. She could arrive at Privet Drive in a blaze of magical beauty and power, and the Dursleys, being the cowards they are, would be too scared to oppose her when she demands Harry come with her.
Well, in some fanfic scenarios, Voldemort might have made a deal with someone who has access to the Dursleys' house. For example, he could have bribed or threatened a neighbor or someone who works in the area. Then, when the time is right, he enters the house under the guise of this person's help and rescues Harry. It could also be that Voldemort has some sort of inside knowledge about the protective spells around the Dursleys' place. He finds a loophole in those spells and slips in to save Harry.
At first, Harry would probably be extremely confused and distrustful. He's been told all his life that Voldemort is the bad guy. But as Voldemort continues to protect him from the Dursleys, Harry might start to question what he knows. He could gradually become more open to listening to Voldemort's side of the story.
Well, perhaps Daphne somehow discovers Harry's awful situation at the Dursleys. She might use magic to break into the house secretly. Then she could use a spell to make the Dursleys forget about Harry so that she can take him away without a fuss.
One thing that could happen is that the wizarding world would be in chaos. The Death Eaters would be confused by Bellatrix's actions, and the Order of the Phoenix would be very suspicious. Harry would be in a difficult position, not knowing whether to trust Bellatrix or not.
Maybe Severus discovers some dark magic plot that involves the Dursleys harming Harry in a more serious way. So, he intervenes just in time. He could use his skills in potions and magic to subdue the Dursleys if they resist, and then take Harry to a safe place, like his own secret hideout or a friendly wizard's home.
Perhaps in this fanfic, Severus doesn't just save Harry physically from the Dursleys. He also saves Harry from the emotional and psychological damage that the Dursleys have been inflicting on him. Severus, having his own traumatic past, might be able to relate to Harry on a deeper level and starts to mentor him in ways that Dumbledore never could. This could lead to a completely different Harry, more confident and less impulsive.
There's also the idea of envy. Voldemort might envy the normal life that Harry could have had if not for the wizarding world's interference. By saving Harry from the Dursleys, he is both taking Harry away from a life he could never have himself and also trying to create a new situation where he can be in control of Harry's life, rather than the Dursleys or the Order of the Phoenix.
It could be part of a plot where Draco is given an assignment by someone in the wizarding world to save Harry from the Dursleys. He initially does it for self - serving reasons like getting in good with certain powerful wizards who are concerned about Harry's well - being. But during the process, he actually starts to care about Harry.
In some fanfictions, Hermione might use her intelligence and magic skills. For example, she could use a powerful spell to create a distraction at the Dursleys' place. Maybe a spell that makes all the dishes in the kitchen start flying around, causing chaos. Then, while the Dursleys are distracted, she could quickly grab Harry and apparate away to a safe place, like the Burrow or a hidden wizarding location she has prepared in advance.