Well, 'The Liberation of Harry' is a good choice. Here, a mysterious force intervenes and saves Harry, leading to a completely different adventure for him. It's a really interesting take on the story.
There are quite a few. One popular one is 'The Protector's Grace' where a powerful wizard steps in to rescue Harry from his miserable life with the Dursleys.
There are fanfictions where a powerful wizard or witch, not part of the main Hogwarts faculty, discovers Harry's situation. Maybe an old hermit wizard who lives on the outskirts of the wizarding world. This wizard uses his magic to create a diversion at the Dursleys' place and then whisks Harry away to his own hidden cottage where he teaches Harry different kinds of magic not taught at Hogwarts.
One popular 'Harry Potter Saved from Dursleys' fanfic could be 'The Early Rescue'. In this fic, perhaps a powerful wizard discovers Harry's situation much earlier and takes him away from the Dursleys. Another might be 'A Different Start' where a magical creature intervenes and whisks Harry to a safe place away from his cruel relatives. There could also be 'Saved by the Unlikely' where a Muggle with some latent magical ability stumbles upon Harry and saves him from the Dursleys.
One common reason is the abuse he endures at the Dursleys'. The writers feel that Harry should not have to put up with such a horrible family. So they create scenarios where he is saved, often by magical creatures or other wizards who can't stand to see a child so mistreated.
One great fanfiction could be 'A New Beginning'. In it, Harry is taken from the Dursleys in a very unexpected way. Maybe a powerful wizard, seeing the mistreatment, decides to rescue him early. Another could be 'The Escape'. It might focus on Harry's journey after being taken from the Dursleys, how he discovers new magic and new friends outside of the Dursley environment. And there's also 'Harry's New Path' which could explore how Harry's character develops differently without the influence of the Dursleys.
Perhaps in the fanfic, a time - traveling wizard goes back to save Harry from the Dursleys. This would lead to a lot of changes in the timeline. Harry might be introduced to magic much earlier and form different relationships. He could also have a different understanding of his parents' history and his own role in the wizarding world as a result of being removed from the Dursleys' environment.
One way could be through the intervention of a powerful wizard or witch who discovers Harry's mistreatment. For example, Dumbledore could send someone like Sirius Black earlier to rescue Harry. Another way might be if Harry himself discovers his magic earlier and uses it to escape. Maybe he finds a magical object in the Dursleys' house that helps him get away from their cruelty.
Hermione Granger could also be a rescuer. She was always intelligent and resourceful. In a fanfiction, she might figure out that Harry was in danger at the Dursleys' through some research or a hunch. Then she could use her knowledge of magic to break into the Dursleys' house and get Harry out. She would be brave enough to take on such a task to save her friend.
Well, the Dursleys were really horrible to Harry. Fans want to see different ways he could have been saved. It also allows for new storylines. For example, if he's saved early, how would his magical development be different? How would his relationships with other characters change? All these possibilities make these fanfics interesting. Also, it gives a chance to explore different magical elements and characters in the context of rescuing Harry from the Dursleys.
Because many fans feel that Harry was mistreated by the Dursleys in the original story. So they create fanfictions where he is taken away from that bad situation. It gives Harry a chance at a different and hopefully better life.
One possible plotline could be that a new character, perhaps a powerful witch or wizard, arrives at the Dursleys' place and whisks Harry away to a magical hideaway. There, Harry meets a love interest who has also been in hiding from the dark forces. They bond over their shared experiences and start to fall in love.