Bungie love causing confusion with how they word things in Destiny 2, and Legendary Lost Sectors are no different. If you have been running around the game trying to find them, that is understandable. Lost Sectors are hidden areas in all the game playable areas 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.
It is not.
Legendary Lost Sectors are actually a lot more like Adventures, and you need to start them in a similar fashion. The first thing you will need to do to get access to Legendary Lost Sectors is to complete some steps in the Raising our Defences quest. You will need to play until you have activated the Seraph Bunker. Simply play through the quest until you reach that point.
When you have the Seraph Bunker activated, 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 it at the start of the season, so be warned.
The mission does take place in a Lost Sector, which is the reason for the name, but that same naming convention makes it a little confusing about how to begin them. Thankfully, now you know, you can just play through the mission, and when the Bunker is up and running you will have access to the Legendary Lost Sector.
Finishing them will get you some much needed Warmind Bits that you can use to upgrade your Seraph Bunker.
