It's a combo of things. A bit of natural physical talent certainly helps, he probably trains like a fucking mad man, his understanding of distance and timing is otherworldly... but his most important and beneficial attribute (by far) is his self belief.
When he says he's the greatest in the world and he's unstoppable, he really, truly believes it. Plenty of people say the same things (especially fighters), but they don't truly, absolutely believe it like Conor does. I honestly think that he has found a way to completely rid himself of self doubt. His confidence is unshakeable. It was particularly noticeable in fights like the Poirier fight. Poirier went through the fighter motions and said all the standard things a fighter says to his opponent, but you could see the self doubt in his face, in his eyes. Conor is not kidding when he says he can literally smell the fear on his opponents, and I'm sure his opponents can truly sense his confidence too. Poirier was saying things to try and intimidate Conor and boost his own confidence... Conor says things because he truly believes them. Although, to be fair, Conor does talk a lot of shit purely to fuck with his opponents head, but if Conor's talking about his own skills and abilities, he believes every word that comes out of his mouth.
You can hear it when he talks and you can see it when he fights. Right from day one he would say "Who da fook is Jose Aldo? He's nuttin. I'm the FW champ." and he believed it. In the Mendes fight, he'd been repeatedly taken down and hit with big shots, but literally as soon as he got back to his feet, he started throwing hard shots again. He wasn't scared of Chad's wrestling at all, because he truly, utterly believed he was the better fighter and knew that he only needed to land the right shots on the feet to win.
Unlike the vast majority of people, Conor is simply not afraid of failure and not afraid to look ridiculous. He's not a basketball player at all, but didn't hesitate for a fucking second to take a shot in MSG with a crowd of people watching... and of course, he nailed the shot. He's also not wary of going to an almost spiritual/supernatural with his fighting. You can see it when he has a staredown. I believe he's actually trying to get in to his opponents soul. Whether this is actually possible in unprovable, but even the attempt unnerves his opponents and bolsters Conor's confidence even further.
All of this is literally the only reason Conor has any kind of chance against Mayweather. When I watched his rant at the boxing media at that fight the other week, I could feel it. The Confidence. It's the first time I thought to myself, "You know what... If anyone's going to do this, it'll be fucking Conor."
I mean, listen to this guy. The level of confidence he has is dangerous.