The Commander Lilith & Fight for Sanctuary DLC pack for Borderlands 2 is a brand new content pack that aims to bridge the story between Borderlands 2 and Borderlands 3, and all owners of Borderlands 2 or Borderlands: The Handsome Collection can download this for free until July 8th.
In order to actually start playing the DLC, boot up Borderlands 2, and load a character that is at least level 30+. If you don't have that, make a new character and opt it to begin at level 30 to be able to get right into the DLC.
Once you are playing on a level 30+ character, go to any Fast Travel station and select the waypoint called "Fight for Sanctury". This will begin the DLC story, and you can return to play it at any time by using this travel station.