Awakening heroes allows us to redistribute the stats of your heroes by consuming nectar. I recommend farming for nectar only via towers as it has the lowest opportunity cost. Awakening heroes can be used to boost either damage or survivability.
I've heard conflicting examples of what is the maximum awakening possible, however most of the players I've spoken to agree that it's 30% of the hero's secondary stat. Eg for my example below using Gabriel, his main stat is agility and his secondary stat is strength. Hence the theoretical maximum to boost damage will be 30% * 27,730 = 8,319. I've been gathering screenshots of maximum awakenings to support this theory from fellow players and I hope to be able to collate more over time.
I've included a video capture from the game to show how to awaken heroes. It's 10min long as I wanted to show a variety of values you can possibly get, as well as how to use some of the other buttons in the awakening interface.