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.
In the fanfic, it could be that Voldemort has developed a new and extremely powerful form of magic that allows him to bypass all the security measures of Azkaban. He may have studied ancient dark magic texts and found a way to nullify the effects of the Dementors and the physical barriers. He then uses this new magic to break into Azkaban and rescue Harry for his own mysterious purposes.