Another reason for winglets: The way that wings produce lift is by creating a low air pressure area on the upper surface of the wing, hence the term 'airfoil.' That much I'm sure you are well aware of. Because of this, there an area of relatively higher air pressure underneath the wing. High pressure flows to low, and the high pressure air tends to curl up over the wingtip to the low pressure area, creating a vortex (as well as drag). This is known as a wingtip vortex, or wake turbulence. Winglets don't eliminate this vortex, they just move the location that it occurs to the top of the wingtip. The theory is that this diminishes some of the drag associated with the vortex by decreasing the size of the vortex itself, thus increasing range. Anyway, that's just how I heard it. You can actually see these vortices when it is humid enough. 757/767s have one coming off the wingtip, as well as the outside edge of the flap (which was what was causing all those crashes behind the 757s)
Click for large version
Photo © George Polfliet
All that aspect ratio stuff might be true, but it doesn't make to much sense to me because all a winglet could do is produce horizontal force, like the vertical stabilizer. And a lot of business jets have winglets, and I'm sure that they don't care about gate clearance.