Well, it could be that Snape overhears some other students gossiping about Harry's odd behavior at home. This could lead him to start paying closer attention to Harry and eventually figure out that he's being abused. For example, if students mention that Harry always seems scared to go back to the Dursleys during holidays.
In some versions, Snape might notice Harry's withdrawn behavior or signs of physical harm during Potions class. Maybe Harry flinches when someone gets too close, which makes Snape suspect something is wrong.
Perhaps Snape has his own reasons to keep an eye on Harry, being related to his mother Lily. One day, he might see Harry with a bruise that couldn't be easily explained away by normal Hogwarts activities. This could prompt him to dig deeper and discover the abuse situation. He might also notice that Harry is always underfed and seems to lack proper care compared to other students, which would be unusual considering he's a wizard at Hogwarts.
In fanfiction, Snape saving Harry often showcases a complex side of Snape's character. It might be a redemption arc for him, as in the original series, Snape had a very ambiguous relationship with Harry. In these fanfictions, it could be due to his love for Lily that he finally decides to fully protect Harry, which is a very interesting take on his character.
They might face magical justice. In some fanfictions, the Ministry of Magic could get involved and strip them of their magic, or at least put them under strict surveillance.
In many fanfictions, Snape might notice Harry's withdrawn behavior. For example, Harry could stop participating in Quidditch, or he might be constantly absent - minded in classes. Snape, being a very observant person, could pick up on these signs. Also, Harry might have some dark circles under his eyes from lack of sleep due to his inner turmoil, and Snape, with his sharp eyes, would not miss such details.
One popular one might be 'Snape's Redemption: Saving Harry'. In this fanfic, Snape discovers the extent of Harry's abuse early on and takes it upon himself to protect Harry from further harm. He uses his position at Hogwarts to keep a close eye on Harry and shields him from the Dursleys' wrath when they try to interfere with Harry's time at school.
Well, it's mainly because it shows a different side of Snape. In the original series, Snape is often seen as a complex and somewhat antagonistic character towards Harry. In these fanfictions, his act of saving Harry gives him a chance at redemption. It also adds an interesting dynamic to their relationship, which was mostly strained in the books.
One possible reason could be Snape's lingering resentment towards James Potter, Harry's father. Snape might see Harry as a reminder of the bullying he endured from James during their school days. So, he takes out his anger on Harry.
In many such fanfictions, Snape might notice Hermione's change in behavior. For example, she could become withdrawn or less enthusiastic in class. He's a very observant character, and her academic performance might also start to decline. He could pick up on the signs during one of his patrols when he sees her looking distressed or if he overhears something that makes him suspect she's in trouble.
Well, Snape saving Harry who he sees as his abused son could start with him seeing the signs that others miss. Snape, being a complex character, might struggle with his own feelings at first. But then he'd use his skills and knowledge to help Harry. He could teach Harry some advanced magic to defend himself. And he might confront Dumbledore about the way Harry has been treated. Eventually, this could lead to a big change in the Hogwarts environment for Harry.
I also like 'When Snape Becomes Harry's Guardian'. In this fanfic, Snape saves Harry from a dangerous situation early on, and then takes on a sort of guardianship role. It has some really interesting interactions between the two characters as they learn to live with each other.
One possible aspect to analyze is Snape's complex character. In many 'Snape Saves Harry' fanfics, it delves deeper into his motives. Snape has always had a love - hate relationship with Harry. In these fanfics, his act of saving Harry might be due to his lingering love for Lily. It shows a different side of Snape, one that is more heroic and self - sacrificing than what we see in the original series at times. This also allows for exploring different plotlines and character development arcs for both Snape and Harry.