My working line of reason is that consciousness is, as he puts it, merely an abstract concept we get attached to and have the illusion of having. There is no consciousness, only the awareness of a single moment--a point in time. What we take for consciousness is nothing more than a succession of these single moments, much as a computer's processor clocks at so many Gigahertz. With each moment of awareness creating new memory and storing that memory back in the brain, the subsequent moments are able to access it.
You coud think that the current moment of awareness is all that ever really "exists", because for all the other previous ones you only have the memory, you have no concrete evidence that it ever existed. There is no past awareness, there is only a current awareness of what we have registered as the past. With that decoupled thinking, you can start to realize that any given moment of awareness that I have is as different (and as similar) from any given moment you have as it is different from a moment I had 5 minutes ago, or ever.
So who I "am" now is as entirely different from who I was just a second ago, in essence, as it would be different from a clone that could be made of me. There is nothing in my flesh ensuring I am a continuous consciousness. There is only the memory. The reason I am closer to my 5-minute younger self than I am to another human being is that the former and I share all the memory up to 5 minutes ago. We have a base commit to our memories in common. But a clone of me, made now, would be exactly as close to my 5-minute younger self.
The bottom line is that I should not think that being teleported should kill me, as that would imply that I'm dying after every moment of awareness, birthing a new person who happens to share all my memories. Indeed I may well be, and we have the illusion that we have a continuous life since we have all these vivid memories of what just happened a second ago, even if it's not us who lived through it.
And though I'm finding this train of thought increasingly alluring, I know nothing of how philosophy ever brushed it, or it may well have tackled it head on and I wouldn't know it.