One reason could be Hermione's excessive worrying. She often frets over small details and Harry might get tired of it and tell her off. For example, if she keeps nagging about the proper way to perform a spell when they are in a hurry during a dangerous situation.
I haven't actually read that specific fanfiction, so I can't summarize it directly. But generally in such a plot, Harry might tell Hermione off due to a misunderstanding. Maybe he thinks she's being too bossy about schoolwork or how they should handle a certain situation in the wizarding world. It could also be related to her over - protectiveness of Ron or some other complex relationship - related issues.
Maybe Hermione has been too pushy with her academic knowledge in the fanfiction. Harry, who might be feeling stressed or frustrated, tells her off for not considering the emotional and practical aspects of a situation. He could say she's being too much of a know - it - all and not listening to others' ideas.
Perhaps a scene where Hermione confronts Snape about his treatment of her friends. In the fanfiction, she could gather evidence of how Snape has been singling out Harry or Ron, and then she barges into his office. She would start off calmly, but as Snape tries to dismiss her, she gets more and more passionate, really telling him off for his behavior.
One possible plot could be that Hermione tells off Molly for being overprotective of Ron. Maybe Molly is constantly interfering in Ron and Hermione's relationship, and Hermione finally has enough and gives her a piece of her mind about respecting their privacy and decisions as adults.
In this fanfiction, it could be that Ginny is involved in some sort of gossip or spreading rumors. Hermione, who values truth and integrity, confronts Ginny about it. She might scold Ginny for being irresponsible with her words and how it can harm others' reputations at Hogwarts.
In some fanfictions, Hermione might be so focused on following the rules that she opposes Harry's more risky but potentially effective plans. Harry, being the more action - oriented one at times, would tell her off for not being more flexible and trusting his instincts.
Hermione's character development in these fanfictions can be complex. When she tells Snape off, she may face consequences that force her to grow in different ways. For instance, Snape could try to retaliate by making her academic life difficult. This would make her learn how to deal with adversity and still stay true to her principles. She might also start to question some of the school's rules and traditions, as she realizes that someone in a position of authority like Snape can misuse them. This leads to her becoming more of a rebel and a reformer within the Hogwarts community, which is a significant development from her initially rule - abiding self.
Yes, there are quite a few fanfictions out there that depict Hermione and Harry kissing. Fanfiction allows for different interpretations of the characters' relationships. Some writers might explore a romantic connection between them that wasn't fully developed in the original series.
In some fanfictions, the use of Parseltongue by Harry could be a way to add an extra layer of mystery and connection between him and Hermione. Maybe he uses it to communicate secretly or to unlock some hidden magic that only Parseltongue can access, which in turn could lead to new adventures for them both.