Harry comes to in the hospital wing and listens to Snape and Fudge in the hallway. Snape tells Fudge that Harry, Hermione, and Ron were Confunded, and he suggests that Harry thinks too highly of himself and has been given too much leeway. Fudge asks about the retreating dementors, but Snape doesn't know why they left Harry, Hermione, and Black alone. At this, Harry opens his eyes and sees Hermione. Madam Pomfrey comes over with a block of chocolate and as Harry tries to get out of bed, she soothingly says that Black will receive the Kiss any moment. Harry's shout of surprise brings Fudge and Snape in from the hallway.
While Snape is clearly angry and prejudiced, he's also not wrong—Harry does think he's above the law and he gets away with plenty of things at school. This starts to suggest that Snape, while misguided, may still be giving out nuggets of truth. However, because Harry won't listen to Snape, he'll never take anything Snape says in this regard seriously.
Harry and Hermione try to tell Fudge that Sirius is innocent and that Peter Pettigrew is an Animagus, but Madam Pomfrey shoves chocolate in Harry's mouth so he stops speaking. Dumbledore arrives and calmly asks to speak with Harry and Hermione. Snape, incensed, asks if his testimony is worthless and screams at Hermione again when she reminds him that he was knocked out and doesn't have the whole story. Dumbledore ushers everyone out of the room and as Snape leaves, he reminds Dumbledore that Sirius proved capable of murder at age sixteen.
When Hermione reminds Snape that he doesn't have the whole story, it shows that she now understands how important it is for a person to have all the information they possibly can before making any important decisions. Snape never saw Pettigrew in his human form, which does mean that he has no reason to believe Lupin and Sirius's tale.
As soon as the door closes, Harry and Hermione start to tell Dumbledore what they know, but he stops them. He reminds them that there's no proof of Sirius's innocence, and Lupin's testimony will count for little given that he's a werewolf. Dumbledore says that he believes them, but he has no power to make Fudge agree. Harry feels as though his last hope is gone as Dumbledore says that they need more time. Hermione's eyes grow round as Dumbledore tells them where to find Sirius and says that they can save two lives if things go well. He reminds Hermione of the law, says he's going to lock them in, and tells Hermione "three turns should do it."
Dumbledore's admission that he doesn't have the power to make Fudge believe their version of events forces Harry to begin to grow up and see the adults in his life as humans, not gods. He also notes that who says something is almost more important than what's said, given that Harry and Hermione aren't believable because they're teenagers and Lupin isn't believable because the wizarding world shuns werewolves.
Hermione pulls a thin gold chain with a tiny hourglass on it out of her robes, calls Harry to her, and throws it around his neck too. She turns the hourglass and things start to blur. Harry finds himself in the entrance hall with Hermione. She drags him into a broom closet and explains that they've gone back three hours in time. She listens to herself, Harry, and Ron shuffle down to Hagrid's hut under the Invisibility Cloak and explains that her necklace is a Time-Turner. McGonagall gave it to her so she could attend her classes, but she had to promise not to tell anyone.
Now that Harry knows the truth about Hermione, many earlier events start to make more sense. In particular, Harry recognizes that while he knew that there was something strange going on, he wasn't especially curious about what was going on with Hermione. This impresses on him that if he wishes to be a good friend going forward, he'll need to be more actively curious about his friends so he can care for them better.
Hermione admits she doesn't understand what Dumbledore wants them to do. They puzzle over what happened three hours ago and Harry realizes that Dumbledore wants them to save Buckbeak so that Sirius can escape on him. He and Hermione run into the forest and sneak around to Hagrid's hut. They watch Hagrid let their earlier selves in and hear Hermione find Scabbers. Harry suggests that they run in and grab Pettigrew, but Hermione tells Harry that they can't change time—people have killed their past or future selves because they think what they’re seeing is dark magic.
When Hermione insists on following the rules of time travel so closely, it suggests that even though they are breaking rules (they are "changing time" by saving Buckbeak, at the very least), there are other rules that are far more important. If Harry were to change what he knows happened at this point, he might not be able to complete his tasks as everything would change.
Harry and Hermione watch Dumbledore, Fudge, Macnair, and the Committee member leave the castle, and see Hagrid escort the trio out his back door. Hagrid lets in the execution party and Harry watches Macnair look at Buckbeak out the window. Fudge explains that he has to read the execution notice and asks Macnair to listen. While Fudge is reading, Harry creeps out and bows to Buckbeak. Buckbeak bows back and Harry starts to drag him towards the forest. Hermione helps Harry drag Buckbeak behind a tree as they hear Hagrid's door open. Macnair is furious and swings his axe into the fence, but Dumbledore sounds amused. Hagrid starts sobbing with happiness.
Because Harry is able to successfully bow to Buckbeak and handle the hippogriff, it suggests that at the very least Hagrid's first class was a success—he successfully taught one student to deal with hippogriffs. This offers some hope for the future that, if Hagrid is able to regain his confidence, he'll be able to become a much better Care of Magical Creatures teacher.
Hermione says that now they'll need to hide until they can fly Buckbeak up to Sirius's window in a few hours. They creep around the edge of the forest until the Whomping Willow is in sight. They watch Ron fight with Scabbers and watch Sirius drag Ron through the roots. Dumbledore walks Macnair, Fudge, and the Committee member back to the castle and minutes later, Lupin lets himself into the tree. Harry suggests that he could go grab the Invisibility Cloak from where they left it, but Hermione stops him just as Hagrid starts walking up to the castle. Minutes later, Snape bursts out of the castle, grabs the Cloak, and vanishes under the tree.
Again, Hermione's insistence on following the rules shows why she was given the Time-Turner in the first place: she still respects rules above all else. She also understands that she and Harry aren't just bound by the rules of time travel; they also have to follow Dumbledore's rules and do only what he asked them to do. Hermione's warning here also saves them from messing up what they know happened in the Shrieking Shack.
As they wait, Hermione asks why the dementors didn't get Sirius. Harry explains what he saw and says that he knows someone conjured a real, powerful Patronus. Slowly and hesitantly, Harry says that it looked like James. Hermione looks worried and Harry turns away. About an hour later, Hermione and Harry watch themselves climb out of the Whomping Willow's roots. Harry grudgingly agrees that they can't go looking for Pettigrew as the cloud reveals the moon. Suddenly, Harry realizes that they have to move in order to avoid the werewolf Lupin. They race back to Hagrid's.
Harry's willingness to tell Hermione the truth about what he saw in regards to the Patronus speaks to the strength of their friendship and the trust between them. This then suggests that they've made up fully and, going forward, they will continue to move towards maturity and engage with each other more like adults than children.
Harry promises not to interfere but goes back outside to keep tabs on events. He hears the dementors swarming himself, Hermione, and Sirius, and decides that he needs to see if James is going to appear. Harry hides behind a bush and watches his feeble Patronus across the lake. Nobody appears and suddenly, Harry understands that he saw himself. He steps from behind the bush and conjures a huge stag Patronus. When it returns to him, Harry whispers "Prongs." A second after it disappears, Hermione and Buckbeak appear behind him. Hermione is distraught, but Harry explains that it's okay since he didn't realize he saw himself. They watch Snape put their past selves on stretchers and take them up to the castle.
When Harry realizes that he actually conjured the Patronus, he learns that there are exceptions to the rules of time travel—but that they can only look like exceptions from the time-traveler's point of view. The fact that Harry was able to mistake himself for James also speaks to the connection that Harry shares with his father, as he's finally able to see what others have been telling him for three years now (that he looks shockingly like his father).
Hermione checks her watch and when they see Macnair leave the castle, presumably to get the dementors, she and Harry get on Buckbeak and fly to the appropriate window. Sirius looks surprised to see them as Hermione unlocks the window. He climbs on behind them and they fly to the top of the West Tower. Harry assures Sirius that they're all going to be fine, and Harry and Hermione yell at him to go.
Successfully saving Sirius means that Hermione and Harry were able to take justice into their own hands and make it happen on a much smaller scale than the Ministry is capable of. This completes their journey towards learning that they can't trust the Ministry to do the right thing.