The moon seems to follow us because it is so far away.
If we compare the traveled distance with the distance between the moon and earth, the traveled distance is very small or negligible so the distance we traveled has no effect on the angle from which we observe the Moon.
This indicates that the moon appears in essentially the same angular position and size, thus, it appears to be moving with us as we move.
Hence, the moon seems to follow because of the constant relative angle.
