My quest for equality began many years ago,
Right here he's telling everyone that he knows what he's doing, he has done this for a long time, is an 'expert' perhaps. Also, it starts right away with an anecdote, a perfect way to showcase 'facts' while grabbing attention.

when i was a boy, my family and i lived in a small farm.
This is Very interesting, he's mentioning a farm, a very noble and humble line of work, but what's more interesting is WHO he's telling this. Republic City has been standing for many years, it's not difficult to believe that a large portion of people in that rally have been born in the city, AKA they have never worked in a farm or anything rural.
This tells them that Amon is more humble than them, that he is from a past 'era'

We weren't rich and none of us were benders
Here he reaffirms his point. He and his family were humble, nothing that gave them the edge in any aspect.

this made us very easy targets for the firebender who extorted my father.
And here he introduces the target. A firebender is the villain in Amon's anecdote, and firebenders created the 100 year war, the freshest war in the mind of the population. Here he also introduces his main argument, benders extort non-benders, an issue we as the audience of the show have seen is very true.

One day my father confronted this man, but when he did, that firebender took my family for me.
Here he reaffirms that his family was noble, his father a righteous man who stood up for himself, and that the firebender was a villain.

Then he took my face. I've been forced to hide behind a mask ever since.
Aaaaand here's the moneymaker, ladies and gentlemen! The thing that stuck out like a sore thumb until i noticed it, the charmcaster, the brainwasher, the single most compassionate thing in the world: Zuko's story.
In recent world developments, there are two heroes: Aang and Zuko. Aang is the Avatar, therefor he's Amon's and the equalists biggest enemy, but Zuko ended the war as well, he was a firebender, but he suffered for that. Zuko also commissioned the Aang statue and he has a statue of himself in the city, for all means, Zuko is a hero.
It makes sense that Amon takes some of Zuko into his story, being scared by a firebending and made ugly, since it's still fresh on people's minds that Zuko suffered and was better for that, so Amon draws parallels between himself and Zuko.

As you know the avatar has recently arrived in Republic City and if she were here she would tell you that bending brings balance to the world, but she is wrong,
No mention of Aang, either. The previous statement brings up a hero from the 100 year war, and this statement omits the other hero, instead he talks about Korra, the Avatar that is still in training. More than that, the Avatar is from a different 'era' and, most importantly, that Korra has the limited view of the Avatar, the classical "bending=balance" thing, but that thought, as he's about to state, is wrong.

the only thing bending has brought to the world is suffering; it has been the cause of every war in every era,
Now we have it, a stronger statement of why bending is wrong, extortion is the modern problem, but not without precedent. What the Avatar says it the means of balance, the humble Amon strikes as the means of war.

but that is about to change. I know you have been wondering what is the revelation, you are about to get your answer.
He promises a change and, to make it stronger, he also states that he's going to reveal what the conference is about, which builds up interest in what's to come.

Since the beginning of time, the spirits have acted as guardians of our world, and they have spoken to me, they say the avatar has failed humanity that is why the spirits have chosen me to usher in a new era of balance,
The entire anecdote builds up to this, if someone tells you that spirits talked to him, you'd say he's wrong. Bush couldn't pull it off, so why is Amon suddenly magical? He's humble. He worked the land, he suffered in the hands of benders and lived with nothing, and humility is always tied with how spirits/gods decide to reward.
There is also a 'confirmation' that the ways of the Avatar are wrong, that the spirits deny and don't believe in the Avatar. So he's saying with a 'backup' that the Avatar is no longer a 'hero' and he is.

they have granted me a power that will make equality a reality. The power to take a person's bending away, permanently.
And here's the strong line, he reaffirms that the Spirits chose him, that his views are the right way and that all of his life and opinions build up to a tangible solution that is open to him alone.