1) Have Battle for Azeroth purchased on your account.
2) Complete the "Insurrection" achievement. This achievement is unlocked once you complete the rather long quest chain in Suramar. Be prepared to spend a least three or four hour on it the first time you do it, because there are lots and lots of quests to do.
3) Be exalted with The Nightfallen. You will get a lot of reputation from doing all the story quests in Suramar, but you will most likely need to do some World Quests as well. Make sure to do the Emissary quests if they are available, and just clear as many World Quests in Suramar as possible, and you will get it soon enough.
You can do follower missions on all your characters as well, and look for missions that give you Nightfall Insignia, and bonus rolls that give you Greater Nightfallen Insignia. These can be mailed to other characters on the same account, so it's' a quick way to get some extra reputation.
The Nightborne recruitment scenarioOnce you have completed the requirements above, head over to Orgrimmar to begin the recruitment scenario at the embassy. It is a short chest chain that ends with a scenario, and it does not take much longer than 15-20 minutes to complete.
Upon delivering the scenario quest, you will unlock the Nightborne as a playable Allied Race on your account, and get the Nightborne Manasaber mount that can be used by all Horde characters.