Who watches the watcher? In your head, there's an apple. You can see it right? Big, bright, red apple. I don't have to do any convincing to tell you that you are not the apple. Your imaginary apple is a "mental object" that you're perceiving, but it isn't conceptually you. You're the thing seeing the apple.
In your head, there's a voice. You know the one I'm talking about, it's the one that's reading these words right now. You can make it talk if you want. Make it say "purple octopus fiddle dee-dop!" See? You're not that voice either. You're the thing hearing the voice.
In your head, there's a movie. You know this one too. It's the one that's playing "that really embarrassing thing" that happened years ago or "the killer response to that argument from earlier." It's not happening right now, it's just a fantasy. When the movie plays, your body reacts, so it really feels like you. But you're not the movie. You're the thing watching the movie.
We might say that our ability to imagine things in our head is a type of "sense" that's distinct from eyesight, touch, hearing, and so on. After all, if I robbed you of your eyesight tomorrow, you could still "see" the apple in your head. We use this thought-sense to "see" the mental objects that are floating around in our minds. But what does the seeing?
In your head, there's a feeling of self. You think "I did such and such thing" or "why is this happening to me". You draw a line between what's you and what's not you. But this is just a feeling. And like the other feelings, it's a mental object that you're perceiving with that thinking-sense. You are not your feeling of self. You're not the watching. You're the thing watching the watcher.
It's like you're looking in a mirror. You look at the mirror and you say, "that's me!" But it's not; it's a mirror.
Your feeling of you may be a natural result of the human body's ability to perceive and create mental objects. If you can perceive mental objects and you create mental objects, then you can create the mental object of "whatever is creating and perceiving the mental object" and then perceive that, and a self arises. It's a recursive loop. 1
The watcher itself is a mental object; the distinction of observed and observer is a mental object. And like the apple, the voice, the movie, and the other feelings, they're arbitrary creations of the mind.
This is, to the best of my understanding2, the theory of "not-self" (the buddhist view), "strange loop self" (the Hofstader view), or "ego death" (the Leary view), depending on who you ask. It's not a nihilistic view; it doesn't say that your physical self does not exist. It says that your perception of self is an emergent property of your ability to think, and is therefore somewhat arbitrary. If you can imagine, you imagine the imaginer. It's "I think, therefore I am not."
If this seems like high-minded philosophy with no practical effect, understand that two of the "bleeding edge" areas of mental health: mindfulness meditation and psychedelic therapy3 have this ego death as the primary active ingredient. Both claim that this isn't just something you can understand intellectually, but something you can physically feel. They imply that the future (and past) of mental health may honestly be taking office workers stressed about their job and blasting them with the fabrication of the ego.
2 If I've gotten this wrong, please let me know!
3 Psychedelics have recently gained much more significant study in therapeutic settings, but larger studies will still need to be done before it can be seriously claimed as the "wonder drug" that it was in the 50s pre-prohibition.