The Sun and Moon both appear to “rise” in the east, and “set” in the west. But as you can see in our video of the Moon’s shadow sweeping across the USA for the 2017 eclipse, you see that the shadow plainly traveled from west to east across the country!
This is a very good question, because it shows how solar eclipses take us out of our normal experience. We are all used to seeing the Sun and Moon rise in the east and set in the west. But if you were situated HIGH above the north pole, the Moon would appear to move counterclockwise (i.e., from your right to left) in its orbit around the Earth. And it is THAT motion of the Moon moving in its orbit, dragging its shadow along behind it, that causes the eclipse path on Earth to move from west to east.
As you watch the total eclipse (you ARE in the path, right?!) you will see the “bite” that the Moon takes out of the Sun start on the “right” side of the Sun’s disk, then slowly grow as the Moon apparently moves across the face of the Sun – from right to left. Since you are looking up at the Sun (with your eclipse glasses on) while facing South, then West is to your right. So as the Moon itself moves from right to left, its shadow also goes from west to east. Here is an example, showing how the eclipse will appear from Cabo:
NASA has also come out with an eclipse animation that shows this PERFECTLY!
During the eclipse, you’re watching the Moon move IN ITS ORBIT, and not as a result of the Earth’s rotation. And that’s pretty cool!