Moody lets Harry in and Fudge greets him happily. Harry notes that he didn't see Maxime the night that Mr. Crouch disappeared, which makes Fudge blush. Dumbledore instructs Harry to wait for him while he shows Fudge the grounds and the three adults leave. Harry greets Fawkes, Dumbledore's phoenix, and sits down. He notices an open cabinet that seems to be glowing. Inside is a shallow basin filled with silvery material. Harry pokes it with his wand and it begins to swirl and reveal an image. Harry notices that he's looking into a room with a chair with chains in the middle. He moves closer to see better and as his nose meets the substance, Harry lurches into the basin and finds himself sitting next to Dumbledore.
When Harry acknowledges what he heard and calls Fudge on his prejudice, it shows that Harry is already developing ways to use his privilege and power to stand up to others in front of other powerful people. Fudge is especially important to do this to, given that he's the one in charge of the entire Wizarding world and, if he so chooses, could make life difficult for people like Hagrid and Maxime if he fears them enough to do so.
Neither Dumbledore nor the other 200 witches and wizards seem to notice Harry. Harry remembers falling into a memory before and to test if this is the same thing, he waves his hand in front of Dumbledore's face. When he gets no reaction, Harry observes the room carefully. The room is arranged to allow everyone to view the chair in the middle. Two dementors escort a much younger Karkaroff into the room and place him in the chair. The chains bind him. Mr. Crouch appears to be in charge and asks Karkaroff for information he promised to share. Harry notices Moody, who mutters that Crouch is going to let Karkaroff out. Nervously, Karkaroff offers several names. Crouch says that they've already caught all but one, but then, Karkaroff names Snape. At this, Dumbledore stands and says that Snape was a Death Eater, but turned spy for Dumbledore.
Moody's muttering and his attitude suggests that he's not convinced that, once a person goes to the Dark side and pledges allegiance to Voldemort, they're capable of ever coming back and rejoining society. It's also important to note that this is the only time that Harry is in the presence of the real Moody, which means that what he sees in the Pensieve is information that the reader can take as truth. The revelation that Snape actually was a Death Eater at one point complicates his backstory and what the trio knows about him, which reminds them that even Snape is a multifaceted individual.
The scene dissolves and changes. Harry finds himself in a different seat and notices a younger Rita Skeeter and an older-looking Mr. Crouch. Harry realizes that this is a different day. Bagman, young and strong, nervously sits down in the chair. Mr. Crouch asks Bagman if he has anything to add in regards to his relationship with the Death Eaters before the council reaches their decision. Mr. Crouch suggests that Bagman spend time in Azkaban for passing information to the Death Eaters, but witches and wizards cry out angrily. Bagman insists he had no idea the person he was speaking to was a Death Eater. Mr. Crouch calls a vote. Nobody votes to put Bagman in Azkaban, but one witch congratulates Bagman on a match last weekend. Mr. Crouch looks furious.
The tenor and the outcome of Bagman's trial suggests that because Bagman is a beloved sports icon for so many, it's unthinkable that he could be working for Voldemort--nobody wants to believe that their idol is evil. The way that Mr. Crouch treats Bagman suggests that he agrees with the Moody from the last scene: Bagman is now fundamentally untrustworthy because he's flirted with evil.
The scene changes again and now, Harry and Dumbledore sit next to Mr. Crouch. The room is silent, save for one small witch crying on the other side of Mr. Crouch. Six dementors usher in four people. One is Barty Crouch, who looks to be in his late teens. He looks terrified. Mr. Crouch looks at the four with hatred and says that they're charged with capturing Frank Longbottom and his wife and torturing them with the Cruciatus Curse. He ignores Barty's cries and sentences the group to life in Azkaban. Barty screams to his mother that he's innocent. Harry hears Dumbledore next to him say that it's time to return to his office and the present Dumbledore pulls Harry out of the memory.
This trial confirms what Sirius told Harry about Mr. Crouch: he hatefully put his own son in Azkaban without much proof of his crimes. While there's no way to know whether this is a final hearing or one that's supposed to encompass the entire process of finding wrongdoing, it's clear that Mr. Crouch hasn't listened to his son and thinks he's unsalvageable, just like Bagman.
Harry apologizes for snooping, but Dumbledore seems unworried. He explains that the bowl is a Pensieve and it allows him to view his thoughts at leisure. To demonstrate, Dumbledore puts his wand to his temple and draws away what looks like a silvery hair. He drops it into the Pensive and Snape appears, saying that "it" is coming back. Dumbledore prods the thoughts again and a scowling girl rises up. She's a young Bertha Jorkins.
The Pensieve allows Dumbledore to read his thoughts in a way that allows him to think critically about them with more ease. This again reminds Harry that he can critically read events just as he can critically read print media.
Harry tells Dumbledore about his dream and Dumbledore reveals that he's been communicating with Sirius and knows that Harry's scar hurt over the summer. Dumbledore explains that he believes that Harry's scar hurts when Voldemort is near or feeling hateful, as he thinks Harry and Voldemort are connected through the scar. Harry says that he didn't see Voldemort in his dream and confirms that there wouldn't be anything to see, since Voldemort doesn't have a body. Harry asks if Voldemort is getting stronger. Heavily, Dumbledore says that last time, Voldemort's rise began with strange disappearances and thus far, there have been three--though most wizards don't read Muggle papers and aren't concerned about Frank Bryce's disappearance.
When Dumbledore admits to reading the Muggle papers, it shows that he understands the value in drawing from a variety of perspectives as he takes in information about the world. It also shows that he's fully aware of how connected the Wizarding world and the Muggle world actually are, as he suggests that Frank Bryce's disappearance is likely related to Voldemort. This shows Harry that though the Wizarding world exists in isolation, it can spill over, especially during difficult times.
Harry hesitantly asks if, in the last trial he saw, they were talking about Neville's parents. Dumbledore asks if Neville hasn't told anyone why his grandmother is raising him and says that Frank Longbottom was an Auror. The Death Eaters tortured him after Voldemort's fall. Both Longbottoms are insane, are in St. Mungo's Hospital, and don't recognize Neville. He says that their testimony was unreliable, given their condition, and Harry asks if this means that Barty Crouch might be innocent. He also asks about Bagman and Snape, and Dumbledore says that neither is a Death Eater. As Harry leaves, Dumbledore asks him to keep the truth about Neville's parents a secret and wishes him luck for the third task.
The fact that the Longbottoms couldn't give reliable testimony again brings up the question of who in society is believable, while Barty Crouch's sentencing despite this unreliable testimony shows that the Wizarding world isn't particularly concerned with truly giving everyone justice. When Dumbledore asks Harry to keep this revelation about Neville's parents secret, he's asking Harry to treat Neville with the empathy that Neville doesn't often receive from his classmates.