Awakening stones are used to increase a stat on your hunter by 1 point.
For example, “White/Grey” -> “Blue” -> “Orange” -> “Purple” (refer to Hunters points)
How do I get Ultimate hunters?
See the video below, you need to raise their stat count above 21 points.
What Happens? Banishing Legendaries & Legendaries in Queue.
You get awakening stones in the mail.