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The basic reason is because the Moon is so far away, but it also has to do with how your eyes and mind work together (it is what is called an optical illusion). If you look at two things (say a building and a tree) that are different distances from you and then walk sideways, you will see that the two objects shift, but that the closer one shifts to the side more than the one that is farther. The Moon is so far away that it does not appear to shift at all. The human mind interprets this in a funny way. It thinks that the Moon is closer than it really is, but that it is moving sideways at the same speed as you are. So you think that the Moon is following you.
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