What is the purpose of a dual LED flash?
The distance from which you can light subjects with a flash is proportional to square root of luminous power due to the inverse square law. A dual LED flash can emit twice as much light as a single LED of the same type, which means you can lit subjects 1.4 times further away. It also draws twice as much power.
A triple LED flash would increase your reach 1.7 times compared to a single LED (1.2 times compared to dual LED). So the difference is not as significant as when stepping from single to dual, but increase of manufacturing costs and power drain is similar. Triple LED flash is not unheard of - Pentax Optio W90 uses three LEDs to aid during macro shots. As they are placed triangularly around the lens, they effectively create a ring light effect.
Generally, if you need more light than a dual flash, you'll be happier with a Xenon flash. A Xenon flash would also let you add regular speedlights off-camera in optical slave mode, but cannot act as a continuous video light / torch.
In iPhone 5s (and hopefully, its successors) the two LEDs are of different color (amber and white). By adjusting their ratio, the camera can adjust color of the flash to match what been determined to be the white balance of ambient light. Traditionally, photographers have had to use gels to obtain similar match between flash and ambient light.
Such color adjustment is intended to reduce washed out look of flash photographs. There are actually two factors contributing to the washed out look - mismatched white balance and on-camera axis of the light. Dual flash fixes only the first of them; axis still remains unnatural for those of us who don't walk around daily with a head lamp (unlike miners). The axis could be fixed by bouncing the flash from a nearby surface, but this is where illumination power falls short for even a dual LED flash, even if there was a reasonable way to direct the light.
I'm so glad guys like you are around for the math part of photography! There is no way I could explain it this well!
Worth noting also that the two LED flashes can be different in colour, as in the iPhone 5s. They refer to this as "True Tone flash", basically intended to reduce the 'washed out' look of people taken with a cheap camera & direct white (cold colour temperature) flash. At a guess they try to estimate the colour temperature of the ambient lighting and match that by varying the ratio of the power between the two LEDs.
@androiddeveloper I'm afraid that would be hard to determine. It was rarely hyped as a feature before iPhone 5s. For example, my Sony-Ericsson K750 (a feature phone from 2005) had dual LED, but specifications usually mention only LED flash.
@Imre Did the iphone 5s have anything special with the dual-LED compared to previous phones that had it?
@androiddeveloper Yes, its LEDs were of different color, as explained in the answer.