The Moon does rotate, exactly once per orbit. That's why we only ever see one face.

It turns once on its axis in the time it takes to go once around Earth, about 27.3 days, so the same side always faces us. If it did not turn at all, or turned at any other rate, we would see different parts of it through the month.

Turns once per orbit: the marked side always faces Earth.

Never turns: over one orbit, Earth would see every side.

How it got that way

Earth's gravity pulls the Moon into a slightly elongated shape. While the young Moon spun faster than it orbited, that bulge was always a little out of line with Earth and always being pulled back into line, and the flexing turned the energy of the spin into heat until the spin slowed to match the orbit. Astronomers call the result synchronous rotation, or tidal locking, and it is common in the solar system rather than a coincidence.

Not quite one face

The orbit is an ellipse, so the Moon's speed around Earth changes through the month while its spin stays steady. For half the orbit it runs ahead of its spin and for the other half behind, so it seems to shake its head, showing a little more of its eastern edge and then of its western one. That rocking, libration, lets us see about 59 percent of the surface over time rather than exactly half.

The far side is not dark

The side we never see gets just as much sunlight as the side we do, a fortnight at a time. It is the far side, not a dark one; at new moon, when the near side is dark to us, the far side is in full sunlight.