GungHo has posted some official details on the awakening special abilities that will become available in version 6.0 that have been previously released. They have also released a monster list for the monsters that will have the awakening abilities. They also showed how the awakening abilities are obtained which is shown below.
So when this update is released there will be a new monster that will appear in the Metadora dungeons and in the regular gacha. When the update is released there will also be a 1 time, 1 run dungeon (similar to the event/samba dungeons) where you can obtain this monster.
This monster is important because you will be able to ‘enhance’ abilities in one of two ways.
The first way is to use this special monster to fuse with the monster you want awoken abilities with, and this will cause these abilities to be awoken.
The second way to enhance abilities is to fuse one of the same monster into your current monster to awake the abilities.
So what is not 100% clear is whether your monster will have to be max level and max skill for this to work (as is shown in this picture of Susanoo). It has been speculated before that they will need to be max level, but not sure on whether max skill is also required (hopefully not).
The release date of the update is still to be determined so stay tuned for further developments.