Well, in some fanfiction scenarios, Snape might rescue Harry from Azkaban because he discovers new evidence that clears Harry's name. Maybe he uncovers a dark plot by some other powerful wizard who framed Harry, and his sense of justice, despite his past animosity towards Harry, drives him to act.
One possible motivation could be guilt. Snape has always had a complicated relationship with Harry. Maybe he feels guilty for not preventing Harry from being wrongly accused and sent to Azkaban. So, he decides to make it right by rescuing him.
Perhaps Snape discovers some dark plot involving the Dursleys that could harm Harry more than he initially thought. The Dursleys were known to be cruel to Harry, and Snape might have found out that they were involved with some Death Eaters or dark magic users who planned to use Harry in a harmful way. So, he decides to rescue Harry to prevent this from happening.
At first, Harry might be extremely suspicious. Snape has always been cold and mean to him in the past. He would wonder what Snape's true motives are. Harry is used to being on his own and not relying on Snape for help, so this sudden rescue would be very unexpected.
In a gay romance fanfiction where Snape rescues Harry, the rescue could be both physically and emotionally. Physically, Snape might save Harry from a dangerous spell or a life - threatening situation. Emotionally, it could be Snape rescuing Harry from feelings of isolation or self - doubt. This could lead to a beautiful relationship where they support and love each other, perhaps defying the norms of the wizarding world.
One possible scenario in such a fanfiction could be that Harry is wrongly accused of a serious crime and sent to Azkaban. Snape, who has always had a complex relationship with Harry, might initially believe in his guilt due to their past animosity. However, as the story unfolds, Snape could start to suspect that there is more to the situation. He might begin his own investigation, secretly at first, to find out the truth behind Harry's imprisonment. This could lead to a series of events where Snape has to face his own prejudices and also deal with the political situation in the wizarding world that allowed Harry to be sent to Azkaban so easily.
One common theme is redemption. Snape, who has had a rather dark past in the original series, through rescuing Harry, finds a path to redemption and love. Another theme is the breaking of barriers. They come from different generations and have had a somewhat antagonistic relationship in the books, but in the fanfiction, the rescue event allows them to break those barriers and form a romantic bond.
In some fanfictions, Snape might rescue Harry out of a sense of duty. Snape has always had a complex relationship with Harry. He could rescue Harry from a Dark Magic attack. Maybe he realizes that Harry is crucial in the fight against Voldemort and despite their personal differences, he steps in to save him. This could also be a turning point in their relationship, leading to a more understanding and less hostile dynamic between them.
There's a fanfiction titled 'A Different Kind of Rescue'. It shows Snape saving Harry in a rather unexpected situation. As the story progresses, their interactions become more frequent and the chemistry between them builds up in a very believable way. It's a great read for those who like this kind of pairing in the Harry Potter universe.