It could be a part of a more complex plot. Voldemort might break Harry out to make the Ministry look bad. By doing this, he can create chaos in the wizarding world and weaken the Ministry's hold on power. Then he can step in and take over more easily. Also, he may have some personal vendetta against Dumbledore and breaking Harry out is a way to get back at him.
Perhaps Voldemort has some sort of prophecy or vision that tells him he needs Harry. He might not fully understand it himself, but he believes that if he breaks Harry out of Azkaban, it will lead to something that benefits him in the long run. Or he could be trying to prove to his Death Eaters that he is the most powerful wizard, by being able to break into Azkaban and rescue someone as famous as Harry.
One reason could be that Voldemort wants to use Harry for his own gain. Maybe he thinks he can turn Harry to the dark side and use his power for some grand evil plan. He might believe that with Harry on his side, he can finally defeat all his enemies and rule the wizarding world completely.
Perhaps Voldemort has some inside help. He could have bribed or threatened one of the guards or even a Ministry official who has access to Azkaban. This person could then help him get in and break Harry out. Or he might have found a secret passage into Azkaban that the Ministry doesn't even know about, and he uses that to his advantage.
Well, perhaps he uses some powerful magic he's learned secretly. Maybe he has help from unexpected allies, like a magical creature that can bypass the prison's defenses.
It could be Sirius Black. Since he has his own experience with Azkaban, he might know some secrets about the place. He could return in some form within the fanfiction and assist Harry in his escape, maybe by leading him through some hidden tunnels or using some special skills he developed during his time in prison.
Maybe he uses some powerful magic spells that he has learned secretly. For example, the Patronus charm could be used to fend off the Dementors which are the main guards in Azkaban. He might also get some unexpected help from his friends or some mysterious characters within the fanfiction world.
One common theme could be power struggle. Since Voldemort is all about power and Harry is his main obstacle, when Voldemort 'breaks' Harry, it shows a shift in the power dynamic.
Perhaps Voldemort has a change of heart. He realizes that Harry has some value to him alive other than just being his enemy.
Well, in some fanfictions, Harry might use some hidden magic he discovers in Azkaban. Maybe he finds a way to turn the Dementors' power against Voldemort. Since Dementors can suck out a person's soul, Harry could somehow direct them towards Voldemort, weakening him enough to deliver the final blow.
One interesting plot point could be how Harry devises the plan to break Sirius out. Maybe he uses his invisibility cloak and some clever spells. Another could be the reactions of the other characters when they find out, like Hermione's concerns about the legality and consequences.
Obviously, Harry and Sirius are the main characters. Harry is the one trying to break Sirius out, and Sirius is the one being broken out.
One possible plot could be that Harry is wrongly accused of a serious crime within the wizarding world and is sent to Azkaban. There, he discovers a dark secret about the prison itself, like there being a hidden force controlling the Dementors. He has to use his skills and the loyalty of his friends to expose this secret and clear his name.