Maybe in the fanfic, there's a mystery that needs to be solved during the school break and Dumbledore believes that Snape and Harry together can figure it out. So he sends Harry to Snape. As they investigate, they face many challenges and through these shared experiences, their relationship evolves from one of animosity to a more positive one.
It could also be a really awkward experience. Snape is used to being strict and cold towards Harry at school. During the break, he might not know how to act around Harry without the school's formal structure. Harry might feel out of place too, but they could slowly build a strange sort of friendship or at least a mutual respect as they are forced to spend time together.
Well, in some fanfictions, Snape might start to see Harry in a different light. Maybe he begins to teach Harry some advanced potions techniques secretly during the summer. He could also grudgingly share some of his knowledge about the Dark Arts in a more controlled way, just to make sure Harry is well - equipped to face future threats.
Their relationship could develop from pure hatred to grudging respect. Harry may come to understand Snape's difficult position in the wizarding world and Snape might see Harry as more than just James Potter's son. For example, Snape could start to appreciate Harry's courage and determination during their time together in the summer.
I don't know who the author of 'harry spends his money fanfic' is. There are so many fanfic authors out there, and without more information, it's impossible to tell.
One possible plot could be that they discover they are brothers early on in Harry's time at Hogwarts. Snape then has to balance his duties as a teacher and his newfound family connection. He tries to protect Harry in secret from the threats within the school, like Malfoy and his cronies.
Perhaps in the fanfic, there is some sort of magic that binds their fates together. Snape might discover this connection and, despite their past differences and his own reserved nature, he decides to propose to Harry as a way to accept this newfound bond. It could also involve Snape trying to make amends for his past actions towards Harry by offering this ultimate commitment.
One possibility could be that Dumbledore arranges for Harry to live with Snape for protection. Snape, though reluctant at first, gradually discovers Harry's hidden strengths and they form an unlikely bond. Harry might learn more about his mother from Snape, which could lead to a deeper understanding between them.
Well, it could be that Snape has some false information that makes him think Harry is in immediate danger from the wrong people at Hogwarts. So, he kidnaps Harry to protect him. Maybe he takes Harry to a secret location like an old, abandoned wizarding house in the Forbidden Forest. There, he tries to figure out what to do next while Harry is really confused and angry at being kidnapped.
In this fanfiction, it might be that Snape sees Harry in a vulnerable state during winter break. Maybe Harry is being bullied by some Slytherins even during the break at Hogwarts. Snape, despite his usual dislike for Harry, steps in to protect him. This action could start to change Harry's perception of Snape, and they could form an unlikely alliance. As time passes, Snape could start to share some of his own experiences with Harry, which helps Harry grow not just as a wizard but also as a person.
The plot could develop with Harry initially being suspicious of Snape's help but gradually realizing Snape's true intentions.