A creature that needs REM sleep but can't enter REM sleep
I imagine some type of creature that needs to enter REM sleep in the same way humans do but can't enter into that state itself. It could have evolved alongside humans and developed a commensal symbiotic relationship (relationship where one benefits and the other is unaffected) with us. The creature could be able to cause humans to enter REM sleep more than they need and feed off this excess. This way the creature is not considered harmful and it would increase its own chance of survival.
If it wasn't small like a mosquito or bacterium then it would probably have to develop a way to cause people to fall more asleep so they don't hear it coming, that ability seems like it would go hand-in-hand with making people enter REM sleep (which is sort of a deeper state of sleep, see more below). Maybe insomniacs would keep them as pets?
You could say it's body has a resistance to the hormones needed to enter REM sleep since it has developed an ability to produce them and spread them to people.
You could easily make it parasitic instead of commensal by making it unable to force people into REM sleep and only able to force them asleep. Now it is stealing the dreams instead of sharing (or instead of stealing while giving them more). Insomniacs may even still keep these as pets, maybe they could be trained to only consume so much REM sleep, or maybe a machine could remove them after you were alseep.
But how is it feeding off it?
As for what it means to feed off of or steal the REM sleep, I am not sure. Maybe it has some sort of way to scan people's brains nearby and "harmonize" with them. This way it can enter into REM sleep itself while its host is. That isn't exactly stealing the REM sleep though. Maybe it would work like a splash in a larger pool making a smaller wave -- because the brain waves are spread out between the two they are less "effective" or something.
Random sleep facts
You have "dreams" the entire night. It's just whatever your brain is thinking about while you sleep. For whatever reason your brain doesn't record these memories and that's why you only remember about five minutes when you wake up, if even that. REM sleep stands for rapid eye movement. Your brain is actually a little more active than "regular" sleep at this time. This is the period when you have the most vivid dreams.
REM sleep is the part of sleep that is important for humans, it's sort of like your brain doing a harddrive disk defrag. When you are in deep sleep, when your brain is the least active, is when you have very slow dreams (ones where you feel like you can't move) and is also when people experience night terrors (only during REM sleep is your body paralyzed, so it is during this period when people toss and turn or scream).
All that to say, REM sleep is really the important part. When you miss a lot of REM sleep your body enters it faster and for longer on subsequent nights.