To earn the "THESE ARE OUR STORIES" trophy in Detroit: Become Human, you will have to spend "Bonus Points" in the Extras section of the game's main menu. You accrue points as you unlock different pathways in each of the game's chapter flowcharts. You will accrue a significant amount of points just through one playthrough, especially if you take the time to investigate as much as possible in a scene and ensure that you don't end up skipping scenes (for example, make sure you find the android during "PARTNERS" so that you get "THE INTERROGATION").
You can spend Points at any time and once you have spent 20,000 in total in the Extras screen you will unlock the trophy. However it might be an idea to not unlock anything until you have 20,000 points in the bank so you can track how close you are to 20,000 as you play.