How Sirius escaped Azkaban

'Er - Mr Black - Sirius?' said Hermione timidly.
Black jumped at being addressed like this and stared at Hermione as though being spoken to politely was something he'd long forgotten.
'If you don't mind me asking, how - how did you get out of Azkaban, if you didn't use Dark Magic?'
'Thank you! Exactly! Precisely what I -'
'I don't know how I did it. I think the only reason I never lost my mind is that I knew I was innocent. That wasn't a happy thought, so the Dementors couldn't suck it out of me... but it kept me sane and knowing who I am... helped me keep my powers... so when it all became... too much... I could transform in my cell... become a dog. Dementors can't see, you now. They feel their way towards people by sensing their emotions... they could tell that my feelings were less - less human, less complex when I was a dog... but they thought, of course, that I was losing my mind like everyone else in there, so it didn't trouble them. But I was weak, very weak, and I had no hope of driving them way form me without a wand...
'But then I saw Peter in that picture... I realised he was at Hogwarts with Harry... perfectly positioned to act, if one hint reached his ears that the Dark Side was gathering strength again...'
'... ready to strike the moment he could be sure of allies... to deliver the last Potter to them. If he gave them Harry, who'd dare say he'd betrayed Lord Voldemort? He'd be welcomed back with honors...
'So you see, I had to do something. I was the only one who knew Peter was still alive...'
'It was as if someone had lit a fire in my head, and the Dementors couldn't destroy it... it wasn't a happy feeling... Dementors couldn't destroy it... it wasn't a happy feeling... it was an obsession... but it gave me strength, it cleared my mind. So, one night when they opened my door to bring food, I slipped past them as a dog... it's so much harder for them to sense animal emotions that they were confused... I was thin, very thin...thin enough to slip through the bars.. I swam as a dog back to the mainland... I journeyed north and slipped into the Hogwarts grounds as a dog... I've been living in the Forest ever since.... except when I come to watch the Quidditch. of course... you fly as well as your father did, Harry...'
'Believe me,' croaked Black. 'Believe me. I never betrayed James and Lily. I would have died before I betrayed them.'

SERVANT OF LORD VOLDEMORT - Pg 395