I've heard that the reason we forget dreams is mostly that since during the REM cycle we are so deeply asleep (we can't even move during REM; that is called Sleep Paralysis), the brain isn't focused on remembering what you are dreaming, but is more like just letting the dream wash over your brain.
When you wake up it may happen that you don't remember anything at all, even though every night you can have from 4 to 7 dreams. You don't necessarily need to wake up during REM sleep to remember your dream, though it does help to remember it with more details.
Anyways, I've also heard that once you wake up if you first before moving a muscle, think of what you dreamed and process it, it helps remembering it. This way you can move some of your dream memories into a place in your brain where it is easier to remember for some more time. Then write down all you can remember in a paper as fast as you can about your dream. Doing this after a few nights can actually help you remember your dreams easier. You even forget 90% of your dream about 5 minutes after waking up.
When you wake up it may happen that you don't remember anything at all, even though every night you can have from 4 to 7 dreams. You don't necessarily need to wake up during REM sleep to remember your dream, though it does help to remember it with more details.
Anyways, I've also heard that once you wake up if you first before moving a muscle, think of what you dreamed and process it, it helps remembering it. This way you can move some of your dream memories into a place in your brain where it is easier to remember for some more time. Then write down all you can remember in a paper as fast as you can about your dream. Doing this after a few nights can actually help you remember your dreams easier. You even forget 90% of your dream about 5 minutes after waking up.