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.
Love for Lily Potter could be a strong motivation. Even though Lily is dead, Snape has always carried a torch for her. He might see rescuing Harry as a way to fulfill a promise he made to Lily, either in his mind or in some past interaction. He realizes that Harry is in a dire situation in Azkaban and that Lily would have wanted him to be safe, so he acts to rescue Harry.
Snape might be motivated by the need to preserve the balance in the wizarding world. With Harry in Azkaban, there could be a power shift that Snape doesn't like. He knows that Harry has certain powers and abilities that are important for the overall good of the wizarding community. So, he takes the risk to rescue Harry to keep things in order.
It could be Dumbledore. He has great power and knowledge. He might have some secret plan all along. He could use his influence and magic to enter Azkaban and free Harry, especially if he discovers that Harry is wrongly imprisoned.
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.
Snape might be motivated by a desire for redemption. He has a complicated past, and saving Harry could be his way of making up for all the bad things he has done or been associated with. Additionally, he might see a bit of himself in Harry's situation, as he also had a difficult childhood, and this empathy drives him to act.
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.
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.
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.
It could be out of loyalty to the wizarding world. Harry is seen as an important figure in the fight against the dark forces. Fleur realizes that Harry's safety is crucial for the future of the wizarding community, so she saves him from the Dursleys who are a threat to his well - being in a different way compared to the dark wizards.
In the fanfic, Snape might use his knowledge of the Dursleys' house layout. He may enter the house unnoticed, find Harry, and then use a Portkey to quickly get them both out of there. He could also have some sort of charm or potion that he uses to counter any magic the Dursleys might have used to keep Harry there, like a spell - breaking potion. And he may have planned the whole rescue very carefully, choosing a time when the Dursleys are least likely to notice, like in the middle of the night.
Well, it could be that when Harry is released from Azkaban, he discovers new magical powers or abilities that he developed during his time there. He might then use these powers to fight against the dark forces that once put him in Azkaban. And he would surely have to deal with the psychological trauma of being in that awful place as he tries to move forward in his life.
He might use the invisibility cloak. Harry could sneak into Azkaban unnoticed and then use magic to break Draco out of his cell. The cloak has served him well in the past, and it would be a simple yet effective way to get past the guards.