/sci/ - Science & Math

In this screenshot of a video it shows how the wave of pressure is visible that was created by the bullet exiting the gun. And it made me think why should it be visible at all?

What I do know is that the difference in pressure in the air causes light to have different speeds. And so because of the fact that light follows a law which says that it always takes the quickest path between two points along its trajectory, that causes the trajectory to bend when the speed changes. But how does that work? Why should light take that quickest path in the first place instead of just going straight through leaving the pressure wave in the air invisible?