One reason could be a prophecy misinterpretation. Voldemort might initially think that Harry is key to his success in a different way than he originally thought, and so he saves him to figure out this new angle. For example, if there was a prophecy about a 'chosen one' and Voldemort realizes he needs Harry alive to fulfill some part of it that benefits him.
One reason could be for comical effect. It's quite funny to think of the powerful and feared Voldemort as a pet. Another reason might be to explore a different power dynamic. If Voldemort is a pet, Harry has a certain level of control over him, which is a complete reversal of their normal relationship in the original story.
One reason could be his pure heart. Harry's pure intentions and lack of desire for power for power's sake gives him an edge. His magic is not corrupted like Voldemort's. It flows more freely and is more in tune with the natural order of magic. So, when he casts spells, they are more effective.
In some fanfiction, Harry leaves to seek more powerful magic. He realizes that the magic in Britain is somewhat limited in the fight against Voldemort. So he travels to other places, like ancient magic - rich regions in Asia or Africa, to find a way to defeat Voldemort. He hopes that by leaving, he can gain the upper hand in the battle.
The wizarding world might be in shock. People who have always seen Voldemort as the ultimate evil would be confused. They would have to re - evaluate their views on Voldemort. Harry, on the other hand, would be in a very strange situation. He might be distrusted by some of his friends who think he has made a deal with Voldemort.
It could be that Harry has lost all trust in the Order of the Phoenix or Dumbledore for some reason. He might think that Voldemort, being the most powerful dark wizard, has the means to protect him from whatever threats he is facing. Also, there could be some false information that leads Harry to believe that Voldemort has a change of heart and can be trusted to protect him.
One reason could be a desire for revenge. Maybe in the fanfiction, Voldemort has done something even more heinous to Harry's family or friends, driving Harry to take extreme measures.
Perhaps Voldemort discovers that there is a greater threat out there. In the fanfic, it could be that another dark force is emerging, and Voldemort sees Harry as an ally of sorts against this new threat. So, he decides to protect Harry in order to deal with this common enemy.
In some fanfictions, Voldemort might save Harry due to a complex change in his character. For example, he could discover a newfound sense of respect for Harry's power or courage. Maybe he realizes that Harry is the only one who can truly understand him, and in a moment of strange kinship, he decides to save Harry from a greater threat that is about to befall them both.
One possible reason could be that Voldemort sees Harry as a symbol of innocence that he wants to corrupt. Just like how he enjoys toying with his enemies, treating Harry like a baby might be a way to belittle him and make him more vulnerable in his eyes.
Some people might create such fanfiction for the shock value. It's a very different and unexpected take on the characters. They want to see how they can twist the traditional story.