As far as I can tell, a kingsmoot is essentially the combination of a political convention and the rap battle scenes from 8 Mile, with the person who has the most clever insults and gets onlookers to chant the loudest ending up as the leader. Euron, friends, is B. Rabbit. After dispatching Theon by patting him on the cheek condescendingly, breaking down what his nephew had messed up Cheddar Bob-style, and using the Greyjoy formerly known as Reek's lack of private parts as the explanation for throwing his support behind a woman, Euron faces off against the much more formidable Yara, aka Papa Doc.
But when she accuses him of murdering her father, Euron flips the script. Rather than deny the charge, he embraces it, and then breaks down all of the reasons killing his brother Balon was a fantastic idea. On Pyke, paying the iron price is the ultimate rationale for any murderous seizure of something that isn't yours, so the others are into this.
And when Yara unveils her own plan to build the biggest fleet the world has ever known, Euron shows even more political savvy. He agrees with his niece's plan, but argues that he should be the one to build and lead it because he’s been all around the world, seen all sorts of cool kinds of nutmeg no one else has, and knows what to do with said fleet. And that's when he breaks down the master plan: sail over to Daenerys and convince her to join forces with them to take the Seven Kingdoms. He also mentions his penis, just to be a jerk to Theon and Yara.
When Yara questions his ability to seduce the Mother of Dragons, he drops some Bismarckian realpolitik. "I'm not going to seduce her," he says. "The Iron fleet will seduce her." And then he drops the ultimate closer: "I wasn't born to be king. I paid the iron price, and here I stand." Yara and Theon might as well have gone to Cranbrook at that point.
Once Euron's won the 'moot, his brother Aeron -- essentially the high Drowned Man of the Ironborn's Drowned God religion, thanks to having nearly drowned himself -- performs the ritual to make a new king, which obviously also involves drowning said king and hoping he comes back to life. But as Euron nearly drowns, Theon and Yara and the people loyal to them escape on their people's best ships and sail off. Euron isn't worried, because crazy people don't worry, so he just makes the reasonable request of building a thousand new ships with the promise of giving the Ironborn the world.
This essentially leaves us with two questions, one immediate and one for the future.