Why does the Moon seem to follow us even if we run away from it at night?

Hello there, the Moon seems to follow us because it is much farther away from us than the things that are around you (example: the cars on the street, or the trees in the park). Since the moon is far, you can't actually observe the movement because the parallax angle is small.
Parallax Angle is the angle made by an object as they move across an observer. Imagine being on a center of a huge protractor. As cars pass in front of you, the angle they make when they move is actually big and they leave your field of view faster. However, for objects far away such as the Moon, the angle they make as they move is small.
Parallax Angles can also be used in determining the distance between us and some other celestial objects far away from us
Reference: http://www2.jpl.nasa.gov/teachers/attachments/parallax.html