And it reconstructs those samples into the original analog wave which is
fully represented, in a continuous manner upon analog reconstruction, in those samples as long as the signal is
bandwidth-limited, as it is in digital (at a frequency above the threshold of human hearing).
See:
Nyquist–Shannon sampling theorem.
There is no discontinuity in the way of "stair steps" or the like, as it is often suggested in false representations of digital.
Here are two very good videos on the subject. It is really worth watching them in their entirety, with an open mind, to gain better understanding: