You don’t form longterm memories while asleep; you lose the last few minutes before you drop off, as well. Short-term memories are held, but they don’t get stored.

And the same applies while you’re dreaming - nothing’s getting recorded. You can pull it out of short-term right after you wake up, but that fades right out.

When you do remember dreams, it’s because you remember remembering them while you were awake. If it was vivid enough to go over hard enough, then the second-hand memory gets stored.

As with most brain questions, nobody knows exactly.

One theory for the purpose of it, is to prevent us from confusing things we dreamed about with things that actually happened.

Edit: Lol, this always gets downvoted, but it’s the truth. We know nothing about the brain; almost all the studies that get reported on are basically this.

Dreams don’t really make any sense. When you wake up, you’re remembering the emotions and linking them with images, but as the feelings fade, unless you were actively making them into a narrative, the random stimulus soup doesn’t have any staying power worth remembering. Trying to remember just corrupts your working memory and will make you change and add details that weren’t there. Same reason eye witness testimony is very often wrong.