I'm sure some of the other guys can give you this HUGE scientific answer. but I can keep it simple. Winglets are used for keeping Drag to a minimum and thus increasing Fuel economy during flight (cruise) which of course can help an airline out in a big way. Hope that helps
Winglets use the fact that there is a flow around the wingtips. Because of the angle of the flow relative to the winglet, the winglet is like a wing at incidence, so it generates lift. The lift vector acts in a slightly forward direction, i.e. it's a small thrust. In addition the winglet helps to reduce the strength of the wingtip vortex which may help to reduce the lift induced drag slightly, but then another one is created at the tip of the winglet.
On the -400 winglets were originally envisioned as a way of extending the wingspan to accommodate the increased MTOW (additional 45,000 lbs). And at the same time allowing the 400 to fit into the same footprint as the 200/300.
During the wind tunnel tests, it was discovered there was an additional benefit of increased fuel efficiency, on the order of 3-4% on the 400.
The new blended winglet technology had gone a step further. These winglets are designed to reduced wingtip vortices, thereby reducing induced drag at cruise speed. The blended winglet improve fuel efficiency by 5-6% on long flights.
Oly720man pretty much summed it up. I suppose I'll bring out the infamous 'pre-fab' winglet explanation I wrote a while ago, anyway, just as a bit of a supplement (to groans, I'm sure, from those of you who have known me long enough ). Anyway:
First, consider a wing: lower pressure on top, higher pressure on the bottom. Simple. It follows easily that air below will want to swirl around the wing tip to the lower-pressure upper surface, as many of you have said already. This spiralling vortex leaving the wing essentially pushes the departing airstream down ("downwash"). This in turn "tips" the lift vector of the aircraft back (because lift is always perpendicular to the departing airstream). That slight rearward tilt of lift creates a component of lift in the streamwise direction -- by definition, drag. That is induced drag.
Now set up a vector. The free airstream flows rearward (towards the back of the airplane), while the vortex flow is approximately parallel to the wing (perpendicular to the rearward flow). It's easy to see that the resultant flow would be at an inward angle to the wing; this is the flow that winglets exploit. Because a winglet is basically a little wing placed at a positive AoA to that resultant flow (the AoA is simply due to the direction of the flow), it generates lift with some forward component, much as propellers do. Because forward lift is really thrust, and thrust is the fundamental force that counteracts drag, drag is effectively reduced. That reason for reduced drag by winglets is accompanied by the fact that they (winglets) also simply block and thus reduce the strength of vortices, reducing drag accordingly.
Winglets aren't just good news, though. Because they create lift, they have all the drag associated with regular wings: parasite drag, induced drag (how ironic, eh?), and interference drag, to name the most basic. Also, under certain flight conditions, winglets will contribute more to drag than they save, making them useless. And, in some cases, drag reduction can be accomplished as or more effectively than winglets simply by extending the wings or adding raked wing tips. Of course, despite all that, winglets usually help performance more than they detract from it, at least for certain types of aircraft and situations...
And to illustrate the various flows involved (very rough and not to scale at all):
...the red is the free airstream, blue is the vortex flow, green is the resultant of those two, and yellow is the resulting lift direction.