Henri Cartier-Bresson described photography as trying to capture the "decisive moment," where all the pieces of a scene come together to be expressive as a whole. An instant before or after just isn't the same and I've missed enough shots by a fraction of a second to teach me to keep my eyes open and my camera on continuous shooting mode.
As a simple illustration of this principle, let's look at the contact sheet of one composition at a recent family gathering. I was originally going to take a picture of my niece playing piano when the scene changed into a girl and her aunt playing together.
And maybe I'm biased, but it helps that she's cute as a button.