1.) They understand each other.
In the episode, the Southern Raiders, it is shown that Aang cannot understand Katara's need for closure and for revenge. Zuko, on the other hand, does. He embraces Katara's darker side and decides to go with her.
Some people say this is why Zutara shouldn't have happened, but I don't think so. Zuko and Katara's anger was honestly righteous, Katara had lost her mom and Zuko did too. That's a lot of pain to carry around, and Aang, as a pacifist can't understand the need for revenge. SPEAKING of which...
2.) They carry each other's pain.
We already know that Zuko has lost his mother and Katara has too. And as shown in the Southern Raiders, Zuko is the one who gets Katara and offers to help her, because he has lost someone important in his life too.
3.) They have the nicest dynamic.
Fire and water, push and pull, yin and yang. Zuko and Katara represent two warring nations, two opposites, and Zutara being canon allows ATLA to show that unity can be brought on.
Zuko loving Katara shows him overcoming the indoctrination and lies fed to him since birth, that the Water Tribes were savage peasants. He learns to leave his nation behind and embrace other cultures more.
Katara loving Zuko shows her overcoming her hate for the Fire Nation, and understanding that she can truly love someone from the nation she thought of as her enemy.
4.) They work well together.
Did you see Sozin's Comet Part 1? The way they faced the rock creatures Toph made? Zuko and Katara were back to back, completely trusting each other and working perfectly. He respects her abilities, and she respects his.
5.) They have similar personality traits.
Both Zuko and Katara are the "parents" of the Gaang. When Sokka climbs into Appa's mouth, it is Zuko who says "get out of the bison's mouth, Sokka" and Katara who says "Sokka, Appa didn't eat Momo. He's probably with Aang." That shows that they are similar and have intersecting beliefs and traits.
6.) They were romantically framed.
Remember the scene when Song tries to touch "Lee's" scar? He doesn't let her. However, the same frame is used when Katara touches Zuko's scar in the Crystal Catacombs, but it's different. Zuko lets her touch his scar, a point of vulnerability for him. More elaboration on that down below...
7.) The Crystal Catacombs
Katara and Zuko were sworn enemies before this scene, but despite this, Katara shows her greatest vulnerability- her survivor's guilt after the death of her mother. Zuko sympathizes with her, and just like that, Katara decides to use her incredibly rare spirit water to heal him.
Think about it. They were literally enemies moments before, but now Katara wants to help him.
She was staring intently at his face, and literally caressing his scar. In return, Zuko had his eyes closed- he trusted her completely, even though he was perfectly aware she could have attacked him at that moment. Zuko trusts her.
In number six, I've already compared them to Song and Zuko- but I want to revisit that. Song doesn't know the real Zuko. She knows Lee, a refugee with a father "fighting in the war." Katara knows the real Zuko- the Fire Nation prince set on hunting her and Aang, but she still chooses to heal and help him.
Even after Zuko betrays them, when he saves Azula when she's losing to Katara, he shoots fire at the water arms, but not at Katara directly, even though that would be more effective.
8.) They have incredible chemistry in only a few moments.
Its true Zutara doesn't have many moments in the show, but the ones I see are enough to convince me. Had Zutara been canon, I'm sure there would be more amazing scenes, and even more meaningful things for these two.
9.) The Final Agni Kai
I cannot even begin to tell you how much this made me feel, and how heavy it is. The fact that Zuko would run faster than lightning to sacrifice himself for Katara is astounding to me.
One might argue that Zuko would take that lightning bolt for anyone, but would Azula shoot it at just anyone? Azula has seen a lot of Zuko and Katara- she's seen Zuko shooting at the water arms attached to her instead of Katara, she knows Zuko's weak spot. Azula is a master strategist.
Azula aimed for Katara because she knew Zuko cared about her to the point of sacrificing himself, so she used that.
Zuko was fighting for the throne of the Fire Nation in that scene, he was fighting for the sake of the world. Letting Katara take the lightning would be most logical, Azula would be vulnerable and him beating her one on one would give him the throne to fulfill his duty. But he just immediately throw that aside because- no, Katara can't die on his watch.
There's actually an amazing analogy for this that I found on Tumblr: Aang would give up the world for Katara, but Zuko gave up himself for Katara.
10.) They have many parallels within the show, most famously Oma and Shu.
After rewatching the Cave of Two Lovers, I began noticing the visual similarities of the Cave and the Crystal Catacombs.
This is no mistake, remember when Shu was wearing blue and Oma was wearing yellow in that story? Well, who was wearing yellow in the Crystal Catacombs?
And guess who was wearing blue?
These parallels were not intentional, which leads me to believe that Katara is the reincarnation of Shu, and Zuko is the reincarnation of Oma.
You know what else makes me think that?
Zuko sacrificing himself for Katara at the Final Agni Kai.
I've already highlighted the importance and weight of that scene. But the fact that Zuko was willing to risk everything, including himself for Katara without hesitation is astounding. I like to believe that since he is the reincarnation of Oma, he will not let his "Shu," now Katara, die, which is why he sacrificed himself so willingly.
That's all my arguments off the top of my head, feel free to agree or disagree with me and debate :)
Just know that I respect your opinion and we can all ship what we want.