In the fanfiction, Stiles may escape through the help of a supernatural being. He could have made a connection with a spirit or another creature before being kidnapped. When he is in distress, this being comes to his aid. Or perhaps he manages to turn Kate's own devices against her. If she has some sort of security system, he figures out how to hack it or use it to create a distraction so he can escape.
He could use his intelligence to manipulate Kate. Maybe he convinces her that he is more useful to her if he is free, and she lets her guard down long enough for him to make a run for it. Also, he could have some hidden skills or knowledge that he uses to break free. Say he knows a spell or a code that unlocks his restraints or the door to his prison.
Stiles usually reacts with a mix of fear and defiance. He's afraid because he's in a dangerous situation, but his strong - willed nature makes him defiant. For example, in many fanfictions, he'll shout at Peter, trying to show that he won't be easily broken.
In this fanfiction scenario, it's likely that Stiles' Omega nature plays a crucial role. Maybe Kate is specifically targeting him because of his unique position in the pack. Stiles could be held in a place filled with various supernatural traps and obstacles. His friends, like Scott and Lydia, would be frantically trying to find him. They might have to face off against Kate's allies as well. As Stiles endures his captivity, he could be learning more about Kate's motives and secrets, which could potentially be used against her when he is rescued.
A good starting point could be a normal day in Beacon Hills. Stiles is going about his business, maybe on his way to meet his friends. Then, suddenly, Kate appears out of nowhere and grabs him. This sudden and unexpected event can immediately hook the readers. You could describe the look of shock on Stiles' face and the cold, determined look in Kate's eyes.
Stiles' father could be a major character. He would be worried sick about his son and use all his resources to try and find him. Maybe he teams up with some of Stiles' friends like Scott or Lydia.
Well, it could be that Kate has some ulterior motives for kidnapping Stiles other than just causing trouble for the pack. Maybe she thinks Stiles has some knowledge about a powerful supernatural artifact or a secret that could help her in her own goals. As the story progresses, Stiles could start to form a strange sort of bond with Kate, not a friendly one but more of an understanding of her twisted mindset, which could add a whole new layer to the plot.
On the other hand, Stiles is also known for his quick thinking. He might start to try and bargain with Peter. He could offer to help Peter in some way if Peter lets him go. For example, if there's a problem in the supernatural community that Peter is facing, Stiles could say he'll use his human ingenuity to solve it. Also, Stiles could be angry. He might start shouting at Peter, calling him out on his actions, and trying to make Peter see how wrong this is.
It could be a rogue werewolf. Maybe this werewolf has a grudge against Stiles' pack and thinks that by kidnapping Stiles, he can get some sort of revenge or leverage.
There's a fanfiction where Stiles is kidnapped by a supernatural being and tortured in a hidden lair. It focuses on his attempts to escape and the friendships that help him through it. The author does an excellent job of building tension throughout the story.
Stiles might initially be shocked. He has been a part of the pack for a long time, and Scott kicking him out would be completely unexpected. He could stand there in disbelief, unable to process what has just happened.
Scott would be extremely worried and immediately start planning a rescue mission. He'd be angry at Theo for targeting Stiles and would use his alpha powers to try and track him down. He might also call on the rest of the pack for help, like Lydia, who could use her banshee powers to sense if Stiles was in danger or where he might be.