I believe that what he is concerned about is that the observer "on top" of the earth can see both the moon in front of him and the sun behind them and that while this is true, the moon has it's shadow on the top relative to the observer, when Tom expects the shadow to be on the bottom relative to the observer.
My diagram is a top-down view of the sun, earth and moon.
The man is standing on the equator, not on the pole.
Also note the red arrow showing the rotation direction of the earth.
Now answering your question:
the Moon seems to rotate from east to west around the earth, so it more or less follows the same movement as the Sun, which also comes from the east and goes to west.
Now, imagine a situation like in the diagram i posted before. The little man is in New Mexico, but instead being the dawn, it is the dusk.
The dusk means that the Sun is moving down towards the horizon.
And as we know now, the moon follows the same east-west movement as the sun (but at different speed), so the moon will also look like moving down into the horizon.
So, in the video, the sun has set some hours ago, and since the moon moves slower, it is also moving down in the horizon, but still is visible after the sun has been set that day.
I hope it's clear now how the thing works.
If not, I would call Diagram Titan to aid us in the explanation.