Perhaps she's being bullied. In the wizarding world, there could be other students who pick on her because she's a 'Mudblood'. This could be really distressing for Hermione, and when Snape finds her crying, he might be conflicted between his general dislike for Gryffindors and his sense of justice.
He might be initially startled. Snape is used to seeing Hermione as a strong - willed and confident student. So seeing her crying could catch him off - guard. But then he might quickly put on his usual cold facade and ask her curtly what's wrong.
Perhaps Hermione is facing some sort of interpersonal issue. She could have had an argument with her friends, like Ron and Harry. In the wizarding world, friendships are crucial. If there's a rift in their relationship, it would deeply affect Hermione. McGonagall, being a perceptive teacher, might then try to mediate and give Hermione advice on how to handle the situation.
Perhaps he's had a falling - out with his friends. In the Slytherin house, there could be a lot of power dynamics and rivalries. If he's been betrayed by someone he thought was a friend, it would be very upsetting.
She might be initially shocked. After all, Draco is usually seen as this tough Slytherin. But then she could feel sympathy. For instance, she might offer him a handkerchief without saying much at first.
It's appealing because it shows a different side of the characters. We usually see Hermione as strong and in control, so seeing her vulnerable is interesting. And Fred, who is often portrayed as the fun - loving prankster, gets to show his caring side. It also allows for new interactions between two characters who don't have a lot of one - on - one time in the original series.
There's this fanfiction called 'Lily's Discovery'. It's quite interesting as it not only focuses on the moment Lily finds Snape crying but also on what led up to that moment. It gives a backstory about Snape's recent struggles in the Slytherin house and how Lily's discovery might be a turning point in their friendship. It's a must - read for fans of their relationship.
One great fanfiction with this scenario is 'A Chance Encounter' where Draco stumbles upon Hermione crying in the library. The story beautifully shows how his initial shock turns into concern and how they start to see each other in a new light. It's a really heart - warming read.
Often, Harry will offer her a handkerchief or a tissue if there is one available. He might also put his arm around her in a comforting gesture. In 'The Common Room Crisis' fanfic, he does just that and then stays with her, listening intently as she pours out her feelings. He is patient and understanding, which is in line with his character in the original books. He tries to make her laugh or at least smile by sharing a joke or a funny memory. This shows his deep friendship and care for Hermione.
One great fanfiction could be 'The Unspoken Pain' where Harry stumbles upon Hermione crying in the library. It delves into the stress she's under from all the studying and the looming threats in the wizarding world. Another is 'A Moment of Weakness' which shows a more personal side of Hermione, perhaps feeling left out by her friends, and Harry is there to comfort her.