The play starts with a big party filled with glitterati – an expression of the political celebrity Sir Robert has become. Party chatter allows the characters to give their two cents on the Chilterns. We learn off the bat that the Lady Chiltern and Sir Robert are "serious," "brilliant," and "of the highest principles."
Mrs. Cheveley busts out her blackmail plan, and the security that Sir Robert felt just two drinks ago evaporates. If he doesn't endorse her crooked investment, she'll tell everyone he got rich quick by selling state secrets as a young man. What if that whole group of socialites downstairs got the news? We can imagine them silently setting down their silver and walking out, or even more fun, tossing oily artichoke hearts at Sir Robert's head.
Now Sir Robert's really stuck between a rock and a hard place – between two strong women who hate each other. On the one side, Mrs. Cheveley threatens public ruin. On the other, Lady Chiltern will revoke her love if Sir Robert does what Mrs. Cheveley says.
We puzzled about this for a moment. The exciting scene in the library looks like a climax. All the physical symbols of the conflict are in play: the letter to Baron Arnheim, the brooch-bracelet Mrs. Cheveley stole, and Lady Chiltern's letter on pink stationary. It feels like a climax – the usually cucumber-like Lord Goring almost physically attacks Lady Cheveley. And it sounds like a climax, the dialogue full of question marks and exclamations points accelerating to the bell that sounds for Phipps.
Then why the puzzlement? Because the protagonist, Sir Robert, is nowhere in sight, and it's customary for the protagonist to be involved in the climax. In this play, however, Lord Goring and Mrs. Cheveley are the most active characters, so it makes sense that the climax takes place between them. Take a look at "Character Roles" for a discussion of Sir Robert as the passive protagonist.
We're back at the Chilterns on the morning of the speech, and we don't quite know what will happen. Because Sir Robert doesn't know that the Arnheim letter (i.e., the source for Mrs. Cheveley's blackmail) has been burned, he could still publicly support the Argentine scheme. He could lose Lady Chiltern. His wife could demand that he rent a cave somewhere for them to live in so he's not tempted by power again.
Because of her decidedly submissive speech of love, we know that Lady Chiltern will have Sir Robert's back, no matter what. She's not even going to make him go hermit. The mini-complication in which Sir Robert refuses to let Lord Goring marry Mabel allows Lady Chiltern to come totally clean and further reaffirm her love.
Wilde serves up the classic comic ending. Marriage!