Although they are closed, during this stage, your eyes moves a lot. Rapid Eye Movement Sleep (REMS) is also known as paradoxical sleep because REMS brain electrical activity is close to the awake one while muscle activity is inhibited.
We usually think about this stage when we talk about dreams because if you were to be woken up during this stage, you would most likely remember a dream quite precisely.
Please note that more and more study tend to show that we also dream during Non Rapid Eye Movement Sleep (NREMS). Yet, we keep poorer memories of such dreams.