Hermione screams as Snape thanks Harry for the Invisibility Cloak. He explains that he took a goblet of the Wolfsbane Potion to Lupin's office, where he found the Marauder's Map. Snape says that it's evidence that Lupin has been helping Black into the castle, and refuses to listen to Lupin trying to explain. He binds Lupin and threatens to kill Black. Nervously, Hermione asks Snape to listen to Lupin, but Snape reminds her that she's already in trouble and then shouts at her to be quiet. Black snarls that if Ron brings Scabbers to the castle nobody will go to Azkaban. Snape threatens to call the dementors and Harry notices a crazy glint in Snape's eye.
Given what Snape knows and how he feels about Lupin and Black, he's able to come to his own conclusions about what's going on here. Notice how Snape screams at Hermione, which stands in sharp contrast to how Lupin handled Hermione's questions. This shows that, especially in intense and emotional moments like this, Snape isn't a teacher that the trio can trust to actually listen to them or think of what's best for them—he's more concerned with his own grudges.
Harry points out that Lupin could've killed him many times over the school year. When Snape refuses to listen, Harry yells that he's pathetic for holding onto a silly grudge. Snape insults Harry and James, then tells Harry to get out of the way. At the same time, Harry, Ron, and Hermione all disarm Snape, which sends him flying back into the wall and knocks him out. Hermione whimpers as Black unties Lupin.
In particular, Hermione's choice to disarm Snape shows that she's learning to think more critically about the systems of power around her and understand that they're not always good or to be trusted.
Lupin tells Harry that it's time to provide proof that his story is true. Black asks for Peter, but Ron weakly asks how Black is sure that Scabbers is Peter. Black pulls out the newspaper clipping of the Weasleys in Egypt, with Scabbers on Ron's shoulder. He explains that Fudge gave him his paper last summer, and Lupin notices that Scabbers is missing a toe. He asks if Peter cut off his finger himself before he transformed. Black affirms this and says that when he cornered Peter on the street, Peter yelled that Black betrayed Lily and James, blew up the street, and ran away with the other rats. Ron won't have it; he insists Scabbers lost the toe in a fight and is so old because his family takes good care of him.
Black's version of events shows that Pettigrew was a master manipulator and fully understood how the Ministry was operating twelve years ago. He knew that because he's an unregistered Animagus, he had an easy escape route and could therefore not ever be around to give any credence to Black's story. This also suggests that Pettigrew understood that Black wasn't going to be given a trial and would therefore have no chance to even try to prove his innocence.
Lupin notes that Scabbers looks awful now, and he and Ron argue about whether Scabbers's health started going downhill after Sirius escaped or after Crookshanks came on the scene. Black says that Crookshanks is a smart creature and has been trying to bring Peter to him, and he stole the list of passwords to get into Gryffindor Tower. Harry thinks this is all absurd as Black says that Peter faked his death by leaving blood on Ron's sheets. This jolts Harry to his senses and he shouts that Black was going to kill Pettigrew like he killed Lily and James. Black agrees that he wants to kill Pettigrew, but Lupin says that Harry has it wrong: Peter betrayed his parents. Harry yells that this isn't true; Black admitted he was the Potters' Secret Keeper.
Because things still don't add up for Harry, he continues to focus his anger on Sirius Black and doesn't yet understand that he doesn't have the whole story. Lupin's presence in this situation likely helps Harry choose to see that Lupin and Black are right, given that Harry already trusts Lupin and seems more willing to continue to do so since Lupin gave back Harry's wand. It's worth noting, however, that Black isn't helping his case by admitting he wants to kill Pettigrew, as that only increases Harry's belief that he's a murderer.
Black says that in a way, he did kill Lily and James: he persuaded them to make Peter their Secret Keeper, not knowing Peter was working with Voldemort. Lupin cuts him off and says that they can prove this by making Peter show himself. He assures Ron that if Scabbers is really just a rat, it won't hurt him, and Ron hands the rat over. As Scabbers squeaks wildly, Lupin and Black point their wands at the rat and it suddenly grows into a shrunken, rodent-like man. Eyes darting to the door, he greets his friends. Lupin casually tells Pettigrew about their chat as Pettigrew insists that Black killed the Potters and now wants to kill him. He says he knew that Sirius was going to escape from Azkaban because Sirius's dark magic is so strong.
Though Scabbers's transformation into an adult man is compelling information enough, Pettigrew's insistence on sticking to the story he's already told shows that he understands the importance of not budging on this matter. He likely believes that if he refuses to admit that he's lying, he still has a chance to convince Harry, Ron, and Hermione that Black is guilty and Lupin is complicit.
Black laughs at the thought of Voldemort teaching him tricks, notices that Pettigrew flinches, and says that Pettigrew has been hiding from Voldemort's supporters, not from him. He suggests they'll be displeased to learn that Pettigrew is alive, and Lupin asks why an innocent man would spend twelve years as a rat. Pettigrew tries to insist that he was just scared of Black, whom he refers to as a spy, but Black growls that Pettigrew has spent his life sucking up to powerful people and he should've seen from the start that Pettigrew was the spy. Harry notices that Pettigrew's face is pale.
The fact that Harry is looking at Pettigrew's body language more than he's listening to what Pettigrew says indicates that he's beginning to shift his thinking to believe Black and Lupin's story. Lupin does have a point that Pettigrew isn't acting innocent, and hasn't acted innocent by spending the last twelve years in his rat form.
Timidly, Hermione asks why Pettigrew hasn't tried to kill Harry already, since he's been sleeping in the same room for years. Black says that Pettigrew doesn't do anything unless there's something in it for him. With Voldemort gone, there's no incentive. Hermione politely asks Black how he got out of Azkaban without dark magic. Black says that the knowledge he was innocent wasn't exactly happy, so the dementors couldn't take it. He spent time as a dog, which made his emotions less complex and kept him sane. When he got the picture of Pettigrew and realized what would happen, it also wasn't a happy thought and he slipped out as a dog and swim to the mainland. He appeals once more to Harry and says he never betrayed Lily and James. Harry nods.
Black's assessment of how Pettigrew operates indicates that Pettigrew has a very different idea of friendship and how it works—namely, it has to give him power, not just camaraderie. Black's explanation of how he got out of Azkaban suggests that a truly innocent person cannot go crazy in Azkaban, as the truth isn't a happy thought; it's simply a fact. This again discredits the Ministry and the dementors as keepers of the peace, as it shows that they're not successful in convicting the right people or punishing them effectively.
At this, Pettigrew falls to his knees and pleads with Black and Lupin. They decide to kill Pettigrew together, so Pettigrew turns to Ron. Ron, however, looks revolted. Pettigrew turns to Hermione and then to Harry, whom he tells that James would've shown mercy. Lupin and Black make Pettigrew confess that he gave Lily and James to Voldemort but as they point their wands at him, Harry leaps in front. Harry says that Pettigrew deserves the dementors, and he doesn't think James would've wanted his friends to be killers. Lupin ties up Pettigrew, they all agree that if he tries to escape they'll kill him, and Lupin splints Ron's leg. Ron and Lupin chain themselves to Pettigrew, while Black levitates Snape's body to take him back to the castle.
By deciding to pardon Pettigrew, Harry momentarily steps into his dad's shoes and in some ways connects the two generations of friendships. In doing so, he also learns that as an individual, he does have the power to enact justice on a small scale where the Ministry cannot be trusted to do so. With Pettigrew alive, Black can prove his innocence—assuming the Ministry of Magic will listen, which they've not proven themselves willing to do thus far.