Moving the story along in Everybody's Gone to the Rapture is one of the chief mechanics of the game. These storyline events can be triggered in a variety of ways.
Trophy
If all events are triggered by the player, there's a special trophy that is awarded called the "Completionist".
Methods
The first method to triggering story elements is simple - look for the golden lights. Remnants of people will show up on the streets and inside interiors as golden shimmering lights, and these are often in pairs. Simply walk up to them, and they will display a story element.
The other story elements are decidedly more complicated to trigger. You will have to click on every single radio, television, and phone in the game to trigger story elements using the "X" button. Radios are somewhat easy to find, as numbers will be repeated and grow louder the closer you get to them. Phones will ring, and you can make it louder the closer you get to it, just like radios. Televisions, however, make no sound, so you will have to manually search them out.
There is no sign that you will have completed all the story triggers in each group except for the radio group, as there is a trophy for radios called "Rapture's Radio Enthusiast".