How can I dream things that are impossible? I remember having dreams whose contents were physically impossible, to the point that when I thought about them afterwards, I couldn't reconcile them with reality in a meaningful way, although when I dreamed them, they seemed to be logical and coherent with reality. The only example I can recall: I was climbing a set of stairs, and there was a large gap in the middle where there were no stairs, and logically I shouldn't have been able to cross from the bottom step to the top, and yet I did. When I think about it afterwards, I can only visualise this as some sort of cheap visual effect, by either teleporting between the two steps, or unnaturally distorting my limbs; but I remember that while dreaming, there was no such effect. It just worked.

How is it that I could dream something that is completely incoherent with reality, or completely impossible, that I can't fully reproduce in my head while awake?
Dreams aren't logical. Your mind is in a relaxed state when you dream. Thoughts become super-imposed and intertwined with each other.

Ideas and imagination is never directly related to interpretation. When you say you imagined stairs, it's actually your idea of stairs. It has no correlation to the real world at all. Your idea of stairs or empty space by themselves have no information related to how objects can physically move from one place to another. Therefore when your mind is being lazy and not focusing, concepts like logical physical movement can be ignored entirely.

We still don't know much about dreams that isn't still speculated about. These are some pretty good places to check out; How Stuff Works - Dreams, Wikipeia - Theories,  The Scientific Study of Dreams