He escapes by changing into his dog form and swimming back to the mainland.
Timeline Notes
Very likely the escape happened within a day or two of Harry's birthday, since it shows up on the morning news on the 31st. Sirius saw the photo of the Weasleys during that previous week, on a copy of the Daily Prophet Fudge had finished reading. This particular photo had appeared a week before Harry's birthday, which would be on the 24th. Fudge wouldn't be carrying a paper more than a day old, so Black would have proabably seen the picture on that actual day. Given reasonable time for all this to occur, it works out logically that he worried frantically about what to do to protect Harry for several days, was heard moaning "He's at Hogwarts" in his sleep as he agonized over the situation, then finally escaped on the 29th or 30th.