Maybe it's related to some past event that hasn't been revealed in the original story. There could be a connection between Snape's family and Hermione's lineage that gives him the right or the need to claim her. For example, if there was a magical debt between their families long ago.
Well, one reason could be that he sees her intelligence and thirst for knowledge, which he respects. He might feel a sort of kinship as he was also a very intelligent student himself when he was younger.
It could be a failed experiment in potions class. Hermione is always eager to try new things in potions, and perhaps this time it backfired, causing her some harm. And Snape would be the one to find her because it's his class.
Maybe she's tired of the often inaccurate portrayals of the characters. Fanfiction sometimes distorts the true nature of Hermione and Snape's relationship as it was in the original books. For example, some fanfictions might make their interactions overly romantic when in fact they were more complex and often adversarial in the canon.
One possible reason could be a misunderstanding. Maybe Hermione thought Snape was involved in some dark plot against her friends. In the wizarding world, there are often secrets and deceptions, and if she got some false information, it could lead her to take such drastic action.
One reason could be differences in teaching methods. Hermione is a very studious and by - the - book student, and Snape has his own rather unorthodox and often harsh teaching ways. For example, in the official books, Snape is known for his sarcastic remarks towards students, and Hermione might not tolerate this in a fanfiction scenario. She could argue with him about how his teaching is not conducive to a positive learning environment.
One reason could be Hermione's intelligence. Snape might recognize that she is smart enough to understand his complex situation. He could see her as someone who could offer a different perspective on his problems.
Hermione is also very confident in her knowledge. Snape is an expert in Potions, but Hermione is no slouch either. So, when Snape tries to undermine her in Potions class, she might respond with sarcasm. For example, if Snape corrects her in a way that she thinks is wrong, she could say 'Of course, Professor, your infinite wisdom must be right' in a sarcastic tone.
In fanfiction, sometimes authors like to break boundaries and explore 'what - if' scenarios. For the 'Snape cheats on Hermione' idea, it could be a way to create drama and conflict. They might be interested in exploring the emotional turmoil it would cause for Hermione, or how it could change the relationships within the wizarding world they've created in their story. It could also be a way to subvert the typical hero - heroine relationships we see in the original series.
One reason could be Snape's knowledge. Hermione, being a knowledge - seeker herself, might feel safe knowing that Snape has a vast understanding of magic, especially the darker aspects. So, she believes he can keep her safe from any threats related to that knowledge.
Some fanfiction writers might do it for shock value. They want to create a very different and extreme situation compared to the original story to attract readers' attention.