The Season 2 finale of Avatar: the Last Airbender took place beneath the vast Earth Kingdom capital of of Ba Sing Se, where Princess Azula claimed victory.
Up until that point, Team Avatar had enjoyed a rough but definite string of successes, from Aang being rescued from the iceberg to the successful defense of the Northern Water Tribe and the recruitment of Toph as his much-needed earthbending teacher. In Ba Sing Se, however, Azula was on the rise, using the manipulative Long Feng and his secret police, the Dai Li, to put Zuko and Katara on the defensive. That drew Aang back to Ba Sing Se, but before he could rescue Katara, Azula struck him down. For the sake of story, it had to be that way.
Aang's New Course
Aang was firmly on the trajectory of "the hero's journey" up until now, facing overwhelming odds and settling his painful past to move forward and accomplish his duty as the avatar. Despite being just 12 years old, Aang had mastered airbending, learned much about waterbending and even had some earthbending under his belt. He was the folk hero of the Water Tribe and the Earth Kingdom by now. At this rate, Aang's journey was a little too easy. Even Azula had struggled to stop him in Season 2, although she came closer than Zuko did thanks to her elite team and superior skills and cunning. If something didn't change soon, the show would grow stale. Protagonists are largely defined by their struggles and setbacks, not by easy victories.
Aang faced his first major defeat at the hands of Azula in the closing scenes of Season 2. This story arc has some parallels to Luke Skywalker getting lured to Cloud City to try and rescue his friends from Darth Vader in The Empire Strikes Back. While training with an ancient master, Aang saw a vision of a friend in trouble, hurried to the scene, then faced defeat and had to be dragged away. While Luke learned of his true heritage, Aang was whisked away and taken to the Fire Nation with his group, and going to the Fire Nation changed everything.
Aang restarted his journey in an unlikely place, and it was there where he learned the true nature of firebending. It is not the element of rage and destruction; it is the element of passion and life. With a more balanced perspective on firebending and the Fire Nation itself, Aang was more receptive to Zuko's overtures of cooperation, and both of them journeyed to the domain of the Sun Warriors later in Season 3. Without these insights and the aid of Zuko, Aang might never have had the power to defeat Ozai. None of this would have ahppened without Azula knocking him out in Ba Sing Se and necessitating his infiltration of the Fire Nation.
Zuko's Slow Metamorphosis
Meanwhile, Azula's victory in the old caverns of Ba Sing Se had an essential, long-term impact on Zuko's life as well. He was already undergoing a personal change to realize his own destiny and his own desires, and he was a man of two minds. Half of him still longed for his father's approval and a place in the Fire Nation royal court, while the other half wanted to stay with Iroh and start a new, peaceful life in the Earth Kingdom (or anywhere outside the Fire Nation). During the battle against Azula, Zuko faced both of these paths when Azula and Iroh both said "join me."
Zuko ultimately chose Azula and the route to the Fire Nation court, which greatly upset Iroh and Katara at the time. But it was the right call, even if it didn't seem that way at first. By joining his sister and returning to the Fire Nation, Zuko immersed himself in the world that he thought he wanted. He lived in opulence as the prince, taking his father's side and spending time with Mai as a lover. But this experience taught Zuko firsthand that this was wrong. He didn't earn it, and he still didn't approve of his father's bloodthirsty, oppressive ways.
Only by joining Azula and helping her win could Zuko realize this; he'd never figure it out by serving tea in Ba Sing Se. Exposure to the current state of the Fire Nation helped galvanize Zuko's will, and he realized that aiding the avatar and restoring balance to the world was his true calling. He could help take down the corrupt Fire Nation nobility from the inside, and launch a new era as the Fire Lord (and a friend to the Avatar).
To help Aang learn the true nature of firebending, and to help Zuko see the true rot of the Fire Nation royal court, it was essential that Azula claimed victory in the caves of Ba Sing Se.