Well, it's also because the Twilight universe has a huge fan base. People who are obsessed with the world of vampires and werewolves in Twilight are eager to read more stories set in that world. Edward, being a central and complex character, offers a lot of room for creativity. Fanfiction writers can explore his thoughts, feelings, and his interactions with other characters in ways that the original story didn't fully cover. For example, they can write about his life before meeting Bella in more detail, or what could have happened if certain events in the story had turned out differently.
Another reason is that fanfiction provides an outlet for fans to engage with the story on a more personal level. They can put their own ideas and interpretations into the fanfiction they write or read. In the case of Edward, fans can create or enjoy stories that show his softer side, his darker side, or his relationships with other characters in a new light. It's a way for fans to keep the Twilight magic alive long after they've finished reading the original books or watching the movies.
The popularity of 'Twilight Edward Story Fanfiction' lies in the fact that fans are already in love with Edward's character from the original series. Fanfiction allows them to explore different scenarios and relationships for him that they might have wished for in the books or movies.