Well, start with research. Play the games, read existing fanfictions (not to copy but to get a feel for what others have done). Decide if you want to write a story that's action - packed, full of mystery, or more about the relationships between the characters. For example, you could write a story set in a post - apocalyptic Hyrule where the rules of the world have changed. Then create a strong protagonist, like a new hero who has to learn from Link's legacy. Plan your story structure and start writing chapter by chapter.
First, you should be really familiar with the Legend of Zelda games. Know the characters, the settings, and the overall story. Then, come up with an interesting idea, like what if Link lost his memory completely and had to start from scratch. Just start writing and let your ideas flow.
To start your loz fanfiction, begin by choosing a specific aspect of the Zelda universe you want to focus on. It could be a particular game's setting or a character. For instance, if you choose the Gerudo Desert from Breath of the Wild. Then think about a unique plot. Maybe a new threat has emerged there that only a young Gerudo warrior can solve with the help of Link, who is passing through. Outline your story a bit and then start writing, developing the characters as you go.