Warning: This is a legacy guide for WoW: Legion. Here is the updated guide to unlocking World Quests in Darkshore at level 120. The procedure is a lot different now.
In order to unlock the World Quests, you need to complete the first part of the War of the Thorns / Burning of Teldrassil story line. This is available to level 110 characters, and you get the first quest to begin the quest chain automatically once you log on to a level 110 character. It will tell you to go to your main capital (Stormwind or Orgrimmar) to begin.
If you don't get this quest automatically once you get to 110, relog to add it to your quest log.
The quest chain itself is actually pretty short, and you can complete it within 20 to 30 minutes if you play somewhat efficient. Just follow the quests until you get to a point where no new quests are available, and at this point the WQs will be available.
The World Quests are on a 24 hour rotation, so you only need to check for World Quests in Darkshore once every day. There are four different World Quests each day, so it does not require a lot of effort to fully clear all the WQs in the entire zone every day.
Each World Quest will give you a piece of ilvl 210+ loot, so it's a very easy way to get decent gear on your alts!