In English atman is usually translated as self and anatman as no-self. However these definitions are problematic. Unfortunately, there are no good English translations of these Sanskrit terms.
To answer your question, we must look at other ways atman can be translated. Atman can mean essence or soul. This represents an unchanging self. No matter what happens in life, atman is unchanging and static. From one lifetime to another, atman is unchanging and static.
In Buddhism, self is seen as conditioned. It lacks essence. It is interdependent on people and things. It is always changing. Therefore it is “not atman” or in Sanskrit, it is anatman.
If you watch your mind during meditation, it is going from one thought to another and one feeling to another. This is the changing self. When your mind becomes quiet, who are you? When your mind is quiet, there is no thought of I.
What gets reborn? The stream of consciousness (vijñāna-santāna) is reborn. But it is not the same consciousness nor is it entirely different. It is evolving. Remember all things are impermanent, even consciousness.
©2016 Stephen L. Martin
Painting: Tempelgärten by Paul Klee