The hexagon corresponds to the latitude of the sharp, pointy, prograde peak at the very top of the above graph, about 78 degrees north. (There's another distinctively shaped band on Saturn, called the Ribbon Wave, that corresponds to the pointy peak at a latitude of about 48 degrees north). That pointy peak is actually one of the smaller ones on the plot, so you might not consider wind speed to be an important factor in driving a unique cloud feature.

Aguiar and her coauthors argue that it's not the wind speeds that are important per se; it's the gradient in wind speeds. Where there are steep contrasts in wind speeds -- adjacent parts of Saturn's atmosphere moving at very different speeds -- you can induce unstable behavior in a fluid, including waves, eddies, and swirls. That little prograde peak in wind speeds at around 78 degrees north is actually the narrowest peak on the graph, so that part of Saturn's atmosphere contains one of the steepest wind speed gradients to be found on the whole planet -- a good place to generate weird atmospheric features, including wavelike disturbances.

In the paper, Aguiar et al. run through a mathematical model that shows how the steep gradient in wind speed can set up a wavelike motion of this high-latitude jet, and that there are likely to be exactly six waves encircling the planet, setting up the hexagon (something that I explain in more detail here), and that wave propagates at about exactly the same speed as the jet flows (meaning that the hexagon will appear nearly stationary with respect to Saturn's rotation). Moreover, they show that the observed wind conditions near the south pole are sufficiently different from those near the north pole that the south pole is not predicted to produce a similar wave, which is good, because there's no hexagon at Saturn's south pole.

Having shown that the idea of wind speed gradients driving the formation of the hexagonal wave, they moved to the laboratory. Fun ensued, and science too.

They set up a cylindrical, rotatable tank 10 centimeters deep and 60 centimeters wide. The tank had a lid and base that were split into concentric sections. They could rotate the inner circle of the lid and floor of the tank at a different rate than they rotated the outer circle of the tank and floor, setting up a gradient in the flow speed of the liquid at the joint between the inner and outer circles. Depending on the relative speeds of the two disks, different things happened. At low relative speeds, there was nothing particularly unusual in the flow, just rotation of the water in the tank. But as the gradient between the two rotating sections was increased, wavelike instabilities started forming at the boundary between the two disks. Depending on conditions, the waves evolved chaotically or sometimes quite stably; there might be as few as two or as many as eight waves encircling the axis of rotation. But for a reasonably wide range of experimental parameters, they produced a wavenumber of 6: a hexagon.