Lair of the Shadow Broker was originally added as the first DLC, and it concerns Liara T’Soni. The Legendary edition already has it, so it doesn’t need to be purchased.
Choice: Complete Shadow Broker or not
Although it can be played at any time, it’s recommended to complete it after the main game, due to some references in there, or at least, you should wait until you’ve exhausted everything you can do with Liara T’Soni in Illium. To start it, talk to Liara, and ask about the Shadow Broker.
Complete Shadow Broker
If you decide to finish the Shadow Broker quest line, you’ll end up with several upgrades which consist of: Assault Rifle Damage, Damage Protection, Heavy Pistol Damage, Shotgun Damage, Sniper Rifle Damage, and Submachine Gun Damage. Also, you’ll get the bonus power Stasis. In terms of story, throughout Mass Effect 3, you’ll hear lots of people references about Liara being the new Shadow Broker, especially since your Mass Effect 2 squad members will know about her being it. You’ll also get to continue your Romance with Liara in it.
Don’t complete Shadow Broker
If you decide to not finish the DLC, you’ll miss out on all the upgrades, and power. In terms of the story, you’ll be completely unaware of Liara being the Shadow Broker, and if you choose not to talk to her about the Normandy, EDI will reveal it to you in a hilarious manner. Also, you won’t be able to continue your Romance with Liara.