Dust, particles of consciousness, is attracted to Adult humans (recall the photograph that Azriel shows the scholars). It is the experience of Adult emotions, particularly eros, or romantic love, that causes Dust to become attracted to a person (and causes other effects such as the settling of their dæmon into a single form). Lyra only learns this when her feelings towards Will change at the end of the last book.
The eros experienced by Will and Lyra is exceptionally strong. Their love brings back balance to the flow of Dust, just as the first "fall" when Eve and Adam discovered love caused Dust to flow towards people in the first place. The uncontrolled flow of Dust (the river in the sky) away from people is a directly a consequence of the rip through the universes caused by the bomb made with Lyra's hair, but indirectly due to the lack of an adult who can love as romantically and strongly as Eve did, and as Lyra does.
The first fall occurred about 33000 years before the novel is set. Prior to that time, humans were animals, and were not aware of their dæmons (as described in Lyra's version of Genesis) This was also the time when the Mulefa first began to ride on seedpods, and they have their own version of the Adam and Eve story.
The prophecy that the witches made was that Lyra was a new Eve, and she must "fall" for Dust to once again flow smoothly down to Adults.