Harry Potter rescues Sirius Black from the tower
Harry Potter rescues Sirius Black from the tower by flying Buckbeak, the Hippogriff, to the window of the office where Sirius is imprisoned, moments before the Dementors are to administer the Kiss. This action, the climax of the night's events, is the culmination of a plan set in motion by Albus Dumbledore and executed by Harry and Hermione Granger using the Time-Turner. It saves an innocent man from a fate worse than death and allows him to escape the Ministry of Magic's custody.
Background and Plan
Earlier that evening, Dumbledore had visited Harry and Hermione in the hospital wing, where they were recovering from their confrontation with Sirius Black, Remus Lupin, and Peter Pettigrew in the Shrieking Shack. Dumbledore, who believed their account of Sirius's innocence, told them that there was no proof to support the story and that the Ministry of Magic, led by Cornelius Fudge, was convinced of Sirius's guilt. He stated that Sirius was locked in Professor Flitwick's office on the seventh floor, the thirteenth window from the right of the West Tower, and that the Dementors would soon perform the Kiss. Dumbledore then gave Hermione a cryptic instruction: "three turns should do it," and locked them in the ward. Hermione revealed the Time-Turner, a device she had been using all year to attend multiple classes, and turned it three times, sending them three hours back in time. Their mission was to save Sirius without being seen.
The Rescue
After hiding in the forest and watching their past selves enter Hagrid's hut, Harry and Hermione waited for the right moment. They saw the execution party arrive for Buckbeak, and as the Committee for the Disposal of Dangerous Creatures read the death sentence inside the cabin, Harry darted out, freed Buckbeak, and led him into the trees. They then waited for the confrontation at the Whomping Willow to unfold, watching as Lupin transformed into a werewolf and Sirius, in his dog form, was dragged into the tunnel. When the coast was clear, Harry and Hermione mounted Buckbeak. Harry urged the Hippogriff into the air, and they flew silently toward the upper floors of the castle. Counting the windows, Harry pulled back on the rope, and Buckbeak slowed to a hover outside the thirteenth window. Harry tapped on the glass. Inside, Sirius Black looked up, his jaw dropping. He hurried to the window but found it locked. Hermione, still gripping Harry's robes, cast "Alohomora!" and the window sprang open.
Escape and Aftermath
Sirius, thin and gaunt, placed his hands on the window frame and heaved himself out, flinging one leg over Buckbeak's back behind Hermione. Harry urged Buckbeak upward to the top of the West Tower, where they landed on the battlements. Harry and Hermione slid off, and Harry told Sirius to go quickly, as Macnair had gone to fetch the Dementors. Sirius, staring down at Harry, said, "How can I ever thank —" but Harry and Hermione shouted together, "GO!" Sirius wheeled Buckbeak around, facing the open sky, and said, "You are — truly your father's son, Harry…" The Hippogriff took off, and they became smaller and smaller until a cloud drifted across the moon and they were gone. Harry and Hermione then raced back to the hospital wing, slipping inside just as Dumbledore locked the door, their presence unnoticed. The rescue succeeded because of the Time-Turner's power to create a closed causal loop: Harry had already seen the stag Patronus that he himself would conjure, and the events he had witnessed from the shore were now being enacted by his past self. The rescue also ensured that the Dementors were removed from the school, as Fudge, furious at Sirius's escape, ordered them packed off back to Azkaban that night.