I would go so far as to say that it is not just the man's "dream" that is important, but all of his dreams.  In fact, dreams end up becoming a theme in this important novel.  In short, the man's dreams always have to do with danger:  past or present.

The man's dreams often accompany a feeling of foreboding because the danger within them isn't specific.  They often involve some imaginary creatures, such as the creature in the cave noted when the story begins.  Although that particular dream is in the present, the man's flashbacks can indicate his dreams from the past.  In this case, the danger can be found in the future.  Why?  Because during the flashbacks, the man's wife is still living and the earth is still fruitful.

The importance of these dreams is explained directly to the boy from the man:

When your dreams are of some world that never was or of some world that never will be and you are happy again then you will have given up.