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 was caught in the crossfire of a duel between other students. Since Hermione is often in the library or around academic areas, she might have been in the wrong place at the wrong time. Snape finding her hurt would be a bit of a shock to him as he knows she's usually more cautious.
Another possibility is a jealous student at Hogwarts. There are always students who might be envious of Hermione's academic achievements. They could set up a prank or a trap that goes wrong and ends up hurting her. For example, a student who is tired of Hermione always getting the top grades in Potions might tamper with her cauldron during a class experiment, resulting in an explosion that hurts her.
Another possibility is that the fanfiction writer wants to explore a different side of their relationships. By having Hermione and Ron hurt Harry, it shows the complexity of human (or in this case, wizard) emotions. It could be a way to show that even the strongest friendships can have cracks and that mistakes can be made. It also allows for character development as Harry has to deal with this new situation, and Hermione and Ron may later realize their mistakes and try to make amends.
He might immediately go into a protective mode. He could use his potions knowledge to quickly make a healing draught for her. After all, Snape is very skilled in potions, and Hermione being hurt would likely trigger his instinct to help.
One possible reason could be the stress from her studies. Hermione is always under a great deal of pressure to excel academically. Maybe she reached a breaking point due to the overwhelming workload. Another reason might be some kind of conflict with her friends. For example, if she had a big fight with Harry or Ron over something serious, it could leave her feeling broken.
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.
One reason could be jealousy. Maybe her friends get jealous of Hermione's intelligence or achievements and act mean towards her in the fanfiction. For example, they might exclude her from certain activities or make snide remarks about her grades.
One possible reason could be that she was attacked by some dark creatures while on a solo adventure in the Forbidden Forest. The forest is full of dangers, and Hermione, being brave and curious, might have ventured too deep and got injured by a creature like a werewolf or a giant spider.
One possible reason could be a misunderstanding during a mission. Maybe Erza's actions were misinterpreted by Natsu, leading to him getting hurt, either physically or emotionally.
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.
One great Hermione and Draco hurt/comfort fanfiction is 'The Green Girl' on FanFiction.net. It really delves into their complex relationship. Draco is often shown as conflicted, and Hermione is there to offer a kind of understanding that he's not used to. It's a beautifully written piece that shows how they can heal each other's emotional wounds.