Your brain doesn't distinguish between "processing" and "storage." In order to remember something, you essentially have to get a lot of the same neurons that fired during the original event to fire again in the same pattern.

When you sleep, your brain is "copying" memories by replaying them while also firing the neurons that will associate those memories in long-term storage, along with associations that need reinforcing. Think of this as indexing and cross-joining the memories.

When your brain does this--replaying snips of memories to associate them--you perceive and experience them: that's dreams. When you wake up, the actual memories have been properly associated and attached, but the many "clips" weren't actually related. Your mind tries to relate things together into narratives, because that's generally how we best recall things. You remember how to navigate to a place by recalling a little story of how you get there. When you wake up, the brain starts trying to put those snips into a sensible order. You start to construct a narrative that links those clips together into a weird story of the dream you had.

Or you don't! If the brain can't find the links, or if you just have other things to think about as you wake up, then the faint echoes of the dream memories fade off and don't cohere into a narrative. That's all our continuity of thought and being really is: echoes.

This same effect is part of why false confessions are pretty easy to get. If you tell someone a good enough story about themselves, and especially if you can provide some evidence, then their mind will start making up appropriate details to fill in the narrative gaps. Your mind really wants there to be a cohesive story so it can link memories up properly.