Bungie likes to confuse the way they phrase things in Destiny 2, and Legendary Lost Sectors are no different. If you've been browsing the game trying to find them, that's understandable. Lost Sectors are hidden areas in all playable areas of the game that contain enemies to kill and loot to find, so it makes sense to think that a Legendary Lost Sector would be a similar thing.
Ce n'est pas.
Legendary Lost Sectors actually look a lot more like adventures, and you have to start them the same way. The first thing you will need to do to gain access to Legendary Lost Sectors is to complete certain stages of the Raising our Defenses quest. You will need to play until you activate the Seraph Bunker. Just play through the quest until you reach this point.
When you have activated the Seraph bunker, you will notice a console to the right of the room. This is where you can launch the legendary Lost Sector. These are very difficult missions, and you may not have enough power to survive them at the start of the season, so be warned.
The mission takes place in a Lost Sector, which is the reason for the name, but that same naming convention makes it a bit confusing how to start them. Fortunately, now you know, you can just complete the mission, and when the Bunker is up and running you will have access to the legendary Lost Sector.
As you finish them, you'll get your much-needed Warmind bits which you can use to upgrade your Seraph Bunker.