"To go north, you must go south. To reach the west, you must go east. To go forward you must go back, and to touch the light you must pass beneath the shadow."
Asshai, Dany thought. She would have me go to Asshai. "Will the Asshai'i give me an army?" she demanded. "Will there be gold for me in Asshai? Will there be ships? What is there in Asshai that I will not find in Qarth?"
"Truth," said the woman in the mask. And bowing, she faded back into the crowd.
Clearly, Quaithe wishes Daenerys to go to Asshai. The first two sentences seem to mean that to reach Westeros in the northwest she must first go to Asshai in the southeast. Touching the light by passing beneath the shadow is more curious, however. Asshai is described as being besides the Shadow, not within it. Why would Dany need to pass beneath the shadow, and what sort of light might she find there?
One theory is that Lightbringer, the Red Sword of Heroes, is hidden within the Shadow and Daenerys must recover it. Another is that Daenerys will perform the unique feat of reaching Westeros by sailing east across the Sunset Sea, thereby landing on the western shores. Martin has emphasized the fact that no one has made the journey before. Finally, the truth Quaithe promises is unknowable. It could be about Lightbringer, it could be about the true danger of the Others,it could be the truth about her family, or it could be all of these things and more besides.
Following remarks from Martin that Asshai will probably only ever be seen in flashback or memory (SSM) -- which seems to be what we see from Melisandre's point of view in A Dance with Dragons -- the notion that Dany will go to Asshai is much less likely. Instead, we believe that the "shadow" she passes beneath is none other than the shadow of Drogon when he lands in Daznak's pit and she confronts and tames him (V: 699). This directly seems to lead to her being put back on the path of destiny on which she seems to have been set from early on. In particular, she realizes what "to go forward, you must go back" means: she travels back into the Dothraki sea, apparently to confront the Dothraki Khal Jhaqo.