It is due to relative motion , we feel something moving.
Suppose, you are going in a travelling bus and another bus also travels side by side with you in the same speed, you feel that you are not moving.
Same is the case of a railway station, when our train is still, but when the train in the next line moves, we feel that our train has started. Isn't it?
If we are taking the distance between moon and us, it's really a great distance and the relative motion is very very small. We can't move that much fast with moon or earth. That's why, we feel moon is also running along with us
I will answer this question in simple and general form rather than in scientific way. The size of moon is big enough to be seen from the Earth. Moon is the nearest object to the earth in the solar system. Thus where ever you go, you can see the moon. You may feel as if it is following us.
