It could be that the Potters knew something about Harry's true magical potential that the Malfoys were suppressing. In the fanfiction world, there could be all kinds of magical secrets. The Potters, seeing Harry's potential being stifled by the Malfoys, decided to kidnap him to allow him to develop his magic freely.
Perhaps the Potters learned that the Malfoys had some sort of magical binding or curse planned for Harry. They might have felt it was their duty as Harry's family (even if not by blood in this fanfiction scenario) to rescue him from such a fate. So they took the drastic step of kidnapping him to keep him safe.
One possible motive could be that the Potters discovered some dark plot by the Malfoys involving Harry. Maybe they found out that the Malfoys were planning to indoctrinate Harry into their pure - blood supremacist ideology against his will. So, to protect Harry from that, they kidnapped him.
Perhaps there's a plot against Harry by some dark forces, and the Potters believe that only they can protect him properly. They might think that the people who have been taking care of Harry so far are not equipped to handle the new and more dangerous threats. Also, it could be that the afterlife in the wizarding world has some rules or opportunities that allow them to try and get Harry back into their family fold. They could have been given a second chance, so to speak, and are eager to take it.
Well, obviously, the relationship between the Potters and the Malfoys becomes extremely hostile. The Malfoys would see the Potters as thieves who stole Harry from them. For Harry, his view of the Potters might be initially one of confusion. He was with the Malfoys, and now he's with the Potters. But over time, he might come to see them as his true family.
Sirius Black could also be part of it. He was Harry's godfather. He might have escaped from Azkaban earlier in this fanfic. He teams up with the Potters to rescue Harry. He could be the one who scouts out the Dursleys' house before the actual rescue operation, using his skills as an Animagus to stay hidden.
One possible reason could be that the twins might have some special abilities or circumstances that draw the Potters' attention more. Maybe they are more outgoing and seem to need more immediate care or guidance. Another reason could be that there is some sort of miscommunication or misunderstanding within the family. For example, the Potters might think Harry is more self - sufficient when in fact he also needs their love and attention.
It could also be that the Potters are under some sort of magical influence or curse. In the fan fiction world, there could be a dark force at play that makes them see the twins as more important. This force might be manipulating their emotions and perception, leading to the neglect of Harry.
In this fanfic, the Potters could have faked their deaths all along. They come back to rescue Harry when they find out how horribly the Dursleys are treating him. They might bring some magical creatures with them to help. For example, a phoenix that can heal Harry if he has any injuries from the Dursleys. Then they take Harry with them on a magical journey to a hidden wizarding community where they can all be safe and happy.
Another possibility is that Harry could find a hidden diary of his parents. The diary could contain their thoughts, experiences at Hogwarts, and even some secrets about the wizarding world that they were involved in. It would give Harry a more personal connection to his parents and their lives before him.
Maybe Dumbledore believes that Harry has some hidden power that is being corrupted by the environment at Hogwarts. So, he kidnaps Harry to a secluded place to train him and help him control this power without any distractions or negative influences from other students or even some of the teachers.
One reason for Blacks to read Harry Potter fanfiction could be the representation (or lack thereof) in the original series. Fanfiction can sometimes address issues like better representation of diverse characters. For Potters, it could be the curiosity of seeing their name in different fictional scenarios within the Harry Potter world. Also, the popularity of the series means there is a vast amount of fanfiction available, and both Blacks and Potters could be attracted to the variety of stories, whether it's about new magical creatures, different relationships between characters, or alternative endings to the main story.
One possible plot twist could be that Snape discovers some hidden magic in Harry that he didn't expect. Maybe a power that's related to the Potters but also has a connection to Snape's own family. Another twist could be that the other Potter family members don't approve of Snape adopting Harry, leading to a family feud within the wizarding world. And perhaps, there could be a dark force that tries to use Snape's new relationship with Harry against them.