Just in the middle of a realization...

Leading up to the escape:

Dementors suck out all the happy memories o thoughts tu have. Sirius knew he was innocent, which wasn't really a happy thought, but it kept him sane while he was in Azkaban.

dulce de azúcar, fudge gave Sirius a newspaper while he was in Azkaban. It happened to be the newspaper with the picture of the Weasley family on it. He noticed that Peter (In the form of Scabbers, of course) was being held por Ron. That's why he was muttering "He's at Hogwarts," remember that?

If you're wondering how any of that is relevant to how he escaped - let me assure you, it is. It's what gave him the motivation to try to escape.

How he escaped:

Dementors do not have eyes, they can feel the presence of a human because of their emotions. Sirius was an animagus, and in the form of a dog his emotions were not as strong. He was so skinny from being in Azkaban for so long, that, in his animagus form, he slipped through the bars and out of Azkaban. He then swam from Azkaban (Azkaban is on an island, I guess?) to shore.

he sneaked past the Dementors in his animagus form and then swam as a dog to the shore. at least that's how I remember it. Sirius did explain it in PoA.

The thought he was locked up while he was innocent was not a happy thought,so the dementors couldn't suck it out of him.That made he didn't lose his mind and keep his magical abilities.Therefore he could change into a animagus,which dementors couldn't know,because they have no eyes.