"Much gasping and exposition ensue"
This is what happens when popular TV shows don't follow popular fan theories or expectations - think about the ending of Dexter, How I Met Your Mother, and Game of Thrones!
Let's look at the meaning of the following words.
Ensue: to happen after something else, especially as a result of it (Cambridge).
Gasp: to take a short, quick breath through the mouth, especially because of surprise, pain, or shock (Cambridge).
- a setting forth of facts, ideas, etc.; detailed explanation.
- writing or speaking that sets forth or explains.
- that part of a play, etc. which reveals what has happened before, who the characters are, etc.
The major revelation that one of Seymour Skinner's men (Armin Tamzarian, a man viewers knew as an upright public servant since S1) had assumed his identity for so long, left the viewers shocked, breathless, and upset. Viewers were riled up.
After the episode aired, viewers began to talk to each other about what had happened, how they were frustrated with the twist/plot, and what it meant going forward (=exposition). You can say that there was a heightened sense of discomfort among viewers after that particular episode. Some were frustrated and angry - enough to merit this line in the article: "...this episode is so widely reviled."
The whole thing can also be applied to the Simpsons universe inside the show - to the characters. In the episode, when the real Seymour Skinner reveals himself in front of the audience, the other characters are shocked/surprised (his mother faints). The impostor gasps, and he is terrified. Allow me to quote user Gary Botnovcan (comments):
"A character we've never seen before [the real Seymour Skinner] says 'You're not Seymour Skinner', and other characters in the room (including the [impostor] we've known as Skinner until now) collectively gasp [and are in shock] . . .
The exposition begins when the new character [the real Seymour Skinner] explains the news by telling us a bit of his own history."