One reason could be that he sees her intelligence and potential. Snape has always had a soft spot for those who are brilliant in potions and magic, and Hermione is top - notch in that regard. He might also recognize her determination and how she's often an outcast among her peers in a way, just like he was. So, he feels a sense of kinship and thus protectiveness towards her.
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.
At first, Hermione might be suspicious of Snape's new - found protectiveness. But as time goes on, she starts to see that he has good intentions. Snape, on the other hand, gradually warms up to Hermione more and more, sharing his knowledge and advice with her.
In the fanfiction, Snape may protect Hermione by shielding her from the Dark Lord's influence. Given his double - agent status, he would be in a position to know if Voldemort has any plans involving Hermione. He could then take steps to keep her out of Voldemort's sight, perhaps by misdirecting Death Eaters or using his own magic to create a protective barrier around her when she's in danger.
One possible reason could be that Snape has shown unexpected kindness to Hermione in the fanfiction. Maybe he has protected her from some danger secretly, which makes her feel she can rely on him and thus feel safe.
In the fanfiction, it might be shown through Snape secretly watching over Hermione during dangerous situations. For example, in a battle or when she's being harassed by other students who are prejudiced against Muggle - borns.
One common scenario could be in the face of other students' bullying. Snape might subtly step in and stop the bullies without making it too obvious. For example, he could give the bullies extra assignments or detention, which would indirectly protect Hermione from their taunts.
In this fanfiction scenario, Snape may become protective of Hermione because she might be involved in some important magical research or discovery. Snape, being a knowledgeable wizard, realizes the significance of her work and decides to protect her from those who might want to stop or steal her ideas. Also, Hermione could be in the middle of a conflict between different wizarding factions, and Snape, for his own reasons, decides to side with her and keep her safe.
Well, in some fanfictions, Snape might be protective of Harry because he sees a lot of Lily in Harry. Lily was Snape's long - loved friend, and Harry is her son. So that connection makes Snape feel an urge to protect Harry from harm, despite their often - rocky relationship in the original story.
Snape might be protective of Draco in fanfiction because in the original series, Snape has a certain loyalty to the Slytherin house. Draco is a young Slytherin, and Snape could see a bit of his own past self in Draco. Also, Snape was tasked with some difficult and dangerous missions by Voldemort. He might want to protect Draco from getting too deeply involved in Voldemort's dark plans to spare the boy from a similar fate as his own.