Why does the darkening of the Moon go from left-to-right during an eclipse?
On any given day, you see the moon rise in the east and (apparently) travel across the sky from your left to the right; therefore, you would assume (incorrectly) that the leading edge of the moon's movement must be to the right. 

That is not correct.

The apparent motion you are seeing is predominantly from the Earth's ~24 hour rotation. But the orbit of the moon is ACTUALLY from west to east as it revolves around the Earth every 29.5 days. Look a the diagram below.

If you were looking down at the earth-moon-sun system far above the ecliptic over the north pole, you would see the Earth spinning in a counter-clockwise direction… and the moon is also revolving around the Earth in a counter-clockwise direction.  

So forget about the Earth's rotation for a moment; it has nothing to do with the moon passing into the Earth's shadow (i.e. the eclipse.) Look at the diagram below, and place yourself on that little ball labeled "Earth" and look into the night sky. Which side of the moon is covered first as it passes behind the shadow cast by the Earth? What do you see?

Left to right.