One way to think of it non-mathematically is like this: Water and air behave differently in this system in how they transport pressure. Water in a hose creates a certain pressure on what’s below it, where you can think of the pressure as measured by the depth of the water. So the weight of 8 vertical inches of water will create 8 inches worth of pressure on whatever’s below it. Air, however, will “teleport” the pressure it’s experiencing from one end to the other end, up or down, without contributing anything significant from its own weight.
So in the outside loop of the hose, on the right side if you’re looking at the video thumbnail, you might have a little 8-inch column of water, which can’t go down any further because there’s air in the way. So it’s exerting 8 inches worth of pressure down on that air. Then the air takes that pressure and “teleports” it to the top of the next coil inwards, so that if there’s 8 more inches of water in the next coil, it’s exerting 16 inches worth of pressure on what’s below it, from the air plus the weight of that much more water. Then the air below it teleports 16 inches of pressure over to the next coil in, and so on for every coil, until the inner coils have as much pressure as they would from a ton of water sitting on top of them, way more than the actual total vertical size of the system.