The player must go to the bell tower and complete the quest Trigger the Gala Event that opens the gates to the Sierra Madre Hotel. You receive the achievement once you have activated the control pad that starts it. This is a story related quest and cannot be missed.
After you have found all three companions and returned them to the central fountain, Elijah will tell you the only way to open and enter to Sierra Madre Casino is to activate the Gala event. This requires you to complete three quests: Fires in the Sky, Strike Up the Band, and Mixed Signals. (These can be completed in any order.) Each quest leaves a companion at a specific point around the Villa. Once all three quests have been completed the quest "Trigger the Gala Event" starts. For this, you just have to activate the Gala event via a console within the bell tower.