The event of a solar eclipse is exactly the same as a large cloud passing in front of the sun above you on a sunny day: you see the shadow of the cloud pass over you across the ground and it becomes momentarily dark at your location. The size of the shadow is also the size of the object above. This proves the moon's diameter is about 70 miles wide. This phenomenon also easily explains why the shadow of the moon during an eclipse travels across the ground from west to east.
The sun causes a shadow for every object on earth as it moves east to west across the sky above the flat plane of earth from dawn until dusk.
Therefore, at sunrise the shadows of a tree or a lamppost are on the WEST side of the object.
At noon, the shadows are directly below the tree or lamppost, as the sun passes directly overhead.
Finally, at sunset the shadows of the tree or lamppost are on the EAST side of the object.
It is this exact same movement of the sun which explains why the shadow of the moon moves from west to east.
The sun is above (at a higher altitude) than the moon. When the eclipse begins, the sun is "rising" above the moon from the east, making the shadow of the moon hit the ground to the west in Oregon as in the August solar eclipse. As the sun passes directly above the moon at its zenith ("noon") point, then the shadow will be directly under the moon, hitting the ground in say, Kansas City. Last, the shadow will be to the east of the moon as the sun "sets" to the west of it, causing the shadow to hit the ground in say, Georgia. This is how the shadow path traveled over 6 hours: Oregon, Missouri, Georgia - WEST to EAST.
The shadow will also be elongated at "rise" and "set," explaining the overall elliptical shape, and more circular at zenith, just like how the shadow of a tree or lamppost will be elongated at rise & set, short at noon.
You can do this experiment for yourself by observing a timelapse photo of the shadow of a lamppost from rise to set.
Here's a good example video: the shadows of these vertical doorposts move across the floor from west to east (left to right) as you see the sun rise in the east (right) side of the camera frame.
This short video demonstrates the exact effect of what happens during the solar eclipse, on a smaller scale: