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.
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.
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.
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 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.
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.
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.
Perhaps Snape uses magic to create a distraction at the Dursleys' place. He could make strange noises or apparitions that scare the Dursleys, and while they are in a panic, he grabs Harry and disapparates. This would be a quick and efficient way for him to rescue Harry.
Maybe Percy uses his magic skills to create a distraction. He could conjure up some strange creatures outside the Dursleys' house. The Dursleys, being afraid of all things magical, would be distracted long enough for Percy to sneak in and get Harry out.
Well, it could be that Draco has a change of heart early on. He realizes that the Dursleys are truly awful to Harry. So he uses magic to create distractions. He could make strange things happen around the house that scare the Dursleys so much that they just want Harry to leave. Then Draco could be waiting outside and whisk Harry away to a safer place, like a hidden place in the wizarding world where he has set up a little haven for Harry.
There are many ways it could be written in fanfiction. Ginny might use her knowledge of the wizarding world and Hogwarts' secrets. She could use a special spell that she learned from an old book in the Hogwarts library. This spell could create a portal right inside the Dursleys' living room that only Harry can see. Ginny could then communicate with Harry through a magical device, tell him about the portal, and he can use it to escape from the Dursleys. Ginny would have to be very careful and plan everything precisely to avoid detection by the Ministry of Magic, as using magic around Muggles without permission is strictly prohibited.