I heard it explained as things that people saw but didn't understand, and a mythology grew up around it. So maybe someone did something that was small or in close and watchers couldn't see it, so they assumed it was some kind of special skill. Or it could be that someone did something that actually didn't make contact, but because most people didn't understand the mechanism, they just ascribed it to "chi", or whatever. I once heard chi defined as anything a 17th century Chinese peasant didn't understand, so there's that...
I personally saw two "no touch" knockdowns done by the late, great Don Angier, who practiced a Japanese sword art. The first was he tipped over a friend of mine who was a former thug and professional kickboxer, just by moving his hand in front of my friend's body. Angier explained it as a matter of understanding people's natural responses and balance to the point where he could manipulate it. Angier wasn't doing it to say how great he was, or claiming to do it in a fight, it was more in the lines of "if you want to be really good at this art, this is the kind of stuff you have to study and understand."
Later that day I saw him Kiai (yell) and cause an attacking black belt to fall on his face. There was context, though. He had just thrown the guy to the floor (hard) four of five times; when the guy went to attack again, he timed his yell so that the guy basically flinched and tripped over his own feet. Angier used his stuff while working as an undercover cop back in the day, and was adamant that he didn't believe in ki. He said it was all just physics and understanding people.