Answer (part 1)
First thing's first. I believe that you may have misinterpreted the audience reaction as being bored of the Jet brainwashing scenes. The audience actually got bored while Azula and the fire nation were attempting to break through the walls of BSS. The hand crank sound effects from the stage crew and the music continued through the frames showing people from the crowd getting sleepy and disinterested. Someone already made a boring joke, so I can no longer say it. However, I can talk about why they may have been bored at this point in time – simply put, the tediousness of waiting for a boring device to fully perforate the wall makes the scene slow and uninteresting, despite the stakes, and compared to, say, when the Uruk Hai used an explosive to invade the Keep in The Two Towers, or the dramatic destruction, in Kingdom of Heaven, of Jerusalem's walls.
Expanded (part 2)
It might be helpful to list out every reaction to conduct a more thorough assessment of what the audience represents:
- clapped when avatar appears
- laughed at avatar saying he's an "incurable prankster"
- responded to momo
- laughter at soka "does this dress make me fat"
- cheered avatar "saving the day" at the NWT
- clapped at toph's sonic wave
- bored while drill is punctuating Ba
- satisfied and impressed with Ty Lee and Mai's performance
- stunned that he gets gravely injured (Azula zaps Avatar)
- cheering for azula temporarily defeating avatar
- laughed at soka's actual jokes given to the actor
- heavy cheering for the battle outcome and azula's triumph over zuko
- standing ovation when the fire lord claims hegemony over all nations
This episode is really quite impressive in how much depth is created in such a compact, small amount of time, and there are several aspects to consider about it.
The first is self-reflection. The way that Aang and the others are portrayed do not really reflect their perception of themselves, which is important, since it makes them self-conscious and fearful about their own confidence in themselves. The fire nation audience laughing or cheering about these eccentricities, defects, or characteristics heightens this effect. More about that to come. Any and all insecurities that they have (such as Zuko's regrets and Aang's attraction to Katara) are put in full display throughout, as it heads to a dramatic conclusion.
Our perspective as viewers is highlighted, in the form of Zuko being stiff in the first half of the series, Aang being a little young and immatured (and thus casting to a girl as boy-characters with conventional plays and opera), Katara crying about hope all the time, and Soka caring a little too much about meat. Fourth-wall breaking occurs when they skip the "Great Divide", often noted as being the worst episode in Avatar history, and well-known to the writers. None of these things are true from a literal standpoint, yet (like Toph says there's some truth in all of it) imply something fundamental in the way the characters are critically viewed.
Of course, there are a few clear inaccuracies, such as a misunderstanding in who the Blue Spirit is, why Iroh and Zuko split up, what Katara really feels toward Jet and Zuko, and how Toph's vision works. We know these things, though it's obvious why the playwrite (Pu-on Tim) would miss that information, and now the audience is slightly more misinformed than they already were going into the play. Also, those things that are still unclear to us, such as Jet's fate, are also unclear in the play. The writers were probably limited in their ability to depict death, and so this is another piece of metacommentary. Tim is not very limited in what he can or can't show, so he has Jet, Zuko, Aang all die, on screen, to live viewers, and there is widespread acceptance of this.
Still, the audience of the play enjoy Aang, Toph, Katara, and Soka even though they're enemies. A comedic relief who's actually helpful and funny? Yes. Aang is still portrayed as free-spirited, and Toph is this hilariously gruff built dude. And then Katara is impassioned to a fault. What's not to enjoy about them? We enjoy interesting and well-designed characters. Them being considered "good" or partisans is irrelevant in this regard in the eyes of the play's audience; they simply want to be entertained, as long as the ending is propitious to the fire nation. Sure, the main cast are characterized in melodramatic ways (so are Azula and the others seen as "protagonists"), and there's a bias seeping into the way that they're being portrayed, but that doesn't mean that the audience members don't find everything having to do with them, or even with side characters such as Jet, of notability.
Although this audience is at first happy to support Aang and everyone in the main cast, their allegiance is ultimately to the fire nation. We see them excitedly applauding Aang beating the fire nation in the State during their invasion of the NWT at the end of S1. This might mean a few things. Perhaps they simply enjoy the carnage and vicarious experience; another interpretation is that Zhao's success isn't as important to the people as say that of the royal family (Azula, Ozai), which parallels how real audiences like Azula as a villain much better than they like Zhao. Something about him doesn't connect with real audiences, probably explaining partially why the writers dropped Zhao as a primary antagonist and moved on to someone they thought would get seen as much more compelling. Even in the play, Zhao is notably absent? Except for a doll which is representing him powerlessly clutched in Aang's hands. Audiences of the play and real audiences like Azula. The first indication that Aang and his friends of Team Avatar #1 are not the protagonists in this stage play, written by and for nationalistic fire nation citizens (or members of loyalist colonies), is when they fervently erupt at Aang being shot with Azula's lightning. Things get worse after that.
The fire nation audience members partially represent the commonalities between fire nation and real audiences of the series; however, they don't break character in that they are still fire nation citizens who are constantly deluded into thinking that their imperialism is justified. This is never as clear as when they give a standing ovation after Ozai declares that, now that the Avatar is defeated, the fire nation is able to complete their goals of, as the people are told, "spreading technological wealth and innovation" to all the other peoples.
Conclusion (part 3)
The episode is really important because not only does it demonstrate the war more clearly through the eyes of the fire nation citizenry, it also provides a summary of everything leading up to the series finale and plays with audience expectation (both in-show and real) of how it will conclude. Tim, the playwrite, obviously has zero idea about how the showdown will go, or if the fire nation will emerge as winners, (except maybe he's also self-deluded into thinking that losing is impossible) and is writing a political show to excite the fire nation base and perpetuate these jingoistic notions of fire nation superiority; although, we know that if any of that happens, it's tantamount to a betrayal of audience trust by the showrunners. It does, however, give a window into the many character fears and insecurities. There is in fact a showdown between Aang and Ozai; Azula vs Zuko. They naturally dread the ultimate, inevitable climax to their conflict, and if they weren't afraid of the outcome, they are now. Aang's been stressing about it for days. Notice how Zuko's face completely crumbles into horror as his actor counterpart is enveloped in stage prop fire. The danger is real now, and ideas of "what if" are creeping into his mind: what if he fails? What is his fate, and how will that translate to global consequences? He's now thinking he must avoid failure, even though there's a real chance that he will get destroyed by Azula, who's (as he knows) a prodigy and a relentless force in fire-bending battle.
That audience's reactions and critical reception are very different, and juxtaposes that of Team Avatar. We sympathize and also have a much different reaction to how the play ends compared to the rest of the play's audience.
The fire audience has zero worries, because they are presented these lies and propaganda on a regular basis. Probably at this point they believe that the fire lord is invulnerable and unimpeachable. They are in some ways a stand-in for real audiences, as human beings with a sense of adventure, wonder, and humor, and in some ways very much the opposite.
