Your diagram is not quite to scale, and the errors are important. Notice that only the hemisphere of the moon which points toward the sun is illuminated, rather than what your drawing shows. This has the following implications.
1) When the moon is new, it rises and sets at the same time as the sun, and is not (mostly) visible at night. The extreme example of this occurs when there is a solar eclipse.
2) When the moon is full, it rises at sunset and sets at sunrise, and is visible during the entire night.
3) When the moon is at half-illumination, it is only visible for half the night.
So the answer to your title question is "We can't."
The dark portion of the moon is not actually dark, since it is illuminated by light reflected by the earth. When the sky is dark, you can see the dimly-illuminated portion fairly clearly, but during daylight hours the sky is brighter than the dark portion of the moon, and that portion is invisible against the sky. That's why you can't see the moon just before and after a solar eclipse.