When your hero reaches level 30, they are not able to gain any more XP until you Awaken them. This increases they level cap by 2 each time, up to 5 times (resulting in a level 40 cap). In order to Awaken your hero you must sacrifice an identical hero each time you do so.
For example, if you have a level 30 Fire Explorer, you will need another Fire Explorer each time you want to Awaken them.
It is not necessary for the hero to be enhanced to +5, such as in evolution. They can be level 0 enhancement.
Heroes that can be acquired through dungeons an indefinite amount of times can be quickly and easily awakened all the way to level 40, since you can simply farm the same hero over and over using skip tickets. This is recommended for beginners, especially with quality 2 star heros such as Water Racing Girl (available on floor 20).