The description of this one is misleading. Fallen Empires aren't given titans when the game starts, nor will they build them to replace fallen ships. They need to become an Awakened Empire (AE) before they'll being making titans. Once an AE has awoken they'll have access to titans, but they won't go past their starting fleet size and therefore won't create any new ships. You or another empire needs to go to war with them and destroy/damage the AE's ship count enough to cause a gap that they can fill with a titan. After that you can send your fleet with a titan in it to destroy their titan fleet. (In my game the achievement popped as soon as the two fleets engaged in combat and before any ship had actually been destroyed)
To make your own titans you'll need to progress through the engineering science branch until mid/end game. The direct prerequisite techs needed are Destroyers, Cruisers, and Battleships after which comes Titans.