The Fathom Dweller is an underwater mount added in Legion, and is tied to a World Quest which you must do various tasks to unlock. The World Quest will award either the Fathom Dweller, or a battle pet called Hungering Claw, which can be sold on the Auction House for a lot of gold!
In order to get started, you need to talk to Drak'thul on the Broken Shores. Speak with him until there is no more dialogue options, then head to a cave located at 58, 54. You need to die in front of the cave, then walk in as a ghost and resurrect. Go to the very end of the cave and look for an object named Mound of Dirt. This can be pretty difficult to spot, so keep looking, because it should always be there. Once you find it, click on it to loot the Weathered Relic.
Now it's time to return to Drak'thul again, and once again you need to talk with him until he does not offer any more dialogue options. You need to close the NPC tab and reopen it a few times before this is done. This will spawn a total of 10 orbs scattered around The Broken Isles, and these need to be clicked on in the correct order to spawn the boss. Below is the correct click order.
1) 38, 37 Cave Aszuna
2) 32,75 Stormheim
3) 41, 80 Val'sharah
4) 32, 72 on Broken Shore Map
5) 59, 13 Cave Aszuna
6) 67, 14 on World Map, under the neutral shark is the cave entrance
7) 56, 38 Highmountain
8) 53, 26 Cave Aszuna
9) 49, 90 on World Map in the sunken ship
10) 37, 71 on Broken Shore Map
The orbs used to have a 50 minute respawn time, which meant a lot of waiting, but this respawn timer has now been reduced to a few seconds, to it should not take a lot of time to get them all anymore.
Upon doing all of this, you will get a World Quest called DANGER: Kosumoth the Hungering. The reward will be either the mount or the pet, and once all of the orbs have been clicked, the boss will spawn. The last orb is conveniently located right next to Drak'thul, so this is where the boss will be.
If you are unlucky and get the pet, you can do all of this again after the raid reset for another chance.