Question

In terms of what you would actually observe, what does it mean to say that the moon apparently moves eastward among the stars?

Answer 1

It means we take a stationery object [Here, the stars] as our frame of reference..
This can also mean that the stars move westward among the moon but then the moon becomes the frame of reference! :)

Answer 2

The synodic period (time from 1 full Moon to the next) is about 30 days.
So if it takes 30 days for one rotation around the Earth, then the Moon moves about 360°/30 or about 12 degrees per day. This would be an eastward motion of about 1 degree every 2 hours. The Moon is about 1/2 degree in size so every hour the Moon moves about 1 Moon diameter eastward with respect to the stars.