Paper Planes is an imaginary story. Most of the things in the song have never happened, however, I was dating a girl back in 2007, so even if the story is imaginary, I have experienced some of the scenes in real life. I used to be extremely unpunctual like the protagonist in the song. So, that’s how I came to imagine the scenario.
The story was built in a way to make the protagonist realize that being late could cost him his girl. As the protagonist reaches the rendezvous point and finds his date missing, he starts feeling anxious. His date is not someone who is ever late. His anxiety makes him imagine various scenarios that could have taken place. He imagines one scenario after another. Each scenario becomes a Paper Plane and starts flying over his head. He imagines so many scenarios that the whole sky is filled with Paper Planes. He imagines the worst-case scenarios and feels how horrible it would feel to lose her.
And by chance, he spots his date sitting on a bench fuming with rage. But the protagonist is extremely relieved that she’s just angry and nothing worse has happened. He apologizes and vows to never be late again.