One possible plot could be that Hermione is wrongly accused of a serious crime against the wizarding world, perhaps a dark magic related offense that she didn't commit. As a result, she is sent to Azkaban. Inside Azkaban, she has to deal with the Dementors and the harsh prison environment. Meanwhile, her friends Harry and Ron believe in her innocence and start to investigate to find the real culprit and clear her name.
A group of Death Eater sympathizers within the Ministry could be the main antagonists. They still hold onto the old pure - blood supremacist ideals and see Hermione, a Muggle - born witch, as a threat. So they fabricate evidence against her and send her to Azkaban to send a message to all Muggle - borns that they are not welcome in the new wizarding order they are trying to create.
One possible plot could be that Hermione discovers a dark secret about Azkaban. And the Ministry of Magic, fearing she'll expose it, sends her there. In Azkaban, she meets some other wrongly imprisoned witches and wizards. Together, they plan an escape while also trying to expose the real criminals in the wizarding world.
Maybe it's a new dark wizard or witch who has infiltrated the Ministry. They manage to manipulate the situation so that Hermione is blamed for their own misdeeds and sent to Azkaban. Their goal could be to cause chaos in the wizarding world and Hermione was the only one who could stop them.
One possible plot could be that Harry is wrongly accused of a serious crime within the wizarding world and is sent to Azkaban. There, he discovers a dark secret about the prison itself, like there being a hidden force controlling the Dementors. He has to use his skills and the loyalty of his friends to expose this secret and clear his name.
It could be a corrupted prison guard. They might be taking advantage of the chaos in Azkaban to carry out their own evil plans, like stealing magic from the prisoners. Harry would have to stop them while also dealing with his own situation in the prison.
Maybe a jealous wizard or witch within the Ministry of Magic. They might be envious of Hermione's popularity and influence among the younger wizards. So they fabricate evidence against her to get her locked up in Azkaban.
Harry might survive in Azkaban because of his connection to his friends, even if they betrayed him. Their memories could give him the strength to hold on. And he may be able to use his magic in unique ways. He could draw on the positive emotions he has left, like his love for his parents, to create a shield against the Dementors. Plus, he may be constantly looking for a way to prove his innocence and that hope could keep him alive.
She could interact with the founders of Hogwarts. Imagine Hermione having a conversation with Rowena Ravenclaw about knowledge and intelligence. It would be a really cool scene as they could exchange ideas about magic and education.
Well, it could be many reasons. Maybe there was a misunderstanding. Some dark magic might have been used to make his friends think he did something bad. Or perhaps a powerful enemy framed him by planting false evidence. It's hard to say exactly without more context from the fanfiction.