I don’t think Sam Harris is trying to say that you don’t actually exist as an individual person, separate from other people. But your idea of your “self”, what you think of as your “self”, is inaccurate. He calls it an illusion.
People have the experience of their “self” as being the thinker or creator of their thoughts. But, really, your sense of “self” is a creation of your thoughts. It is just made of thoughts itself. So it cannot be the creator of the thoughts.
Your sense of “self” is just one small part of your mind: it is not your entire mind, it is not the creator of your mind, it is not in control of your mind. It is a creation of your mind, not your mind itself.
Understanding this, or being able to think about it in this way, can be helpful to people. Seeing your “self” as a creation of your mind can make it easier to break out of old habits, easier to change and be a different type of person. The idea you have of who you are is just an idea (or, a group of ideas): it is not who you are.
(Btw, Sam also applies this stuff to free will – I think he believes the non-existence of free will follows from these ideas about self being an illusion – and I disagree with him on that part.)