How are double rainbows formed and why the colours are inverted in a double rainbow?

All rainbow require the presence of the sun and rain, in order to see a rainbow. The sun must be to the viewers back, and rain must be falling ahead of the viewer. Rainbow will only occur at exact angle of 48 degrees, anything less than that, then light will simply pass through. As sunlight breaks through the clouds and beams towards the raindrops, some of the light bends-this process is called "refraction". This refraction causes the light to separate in different colors. The refracted light waves then bounce-or being reflected again-off the circular edge of the raindrop, and then they refract again as they exit the raindrop and travel through air. Because raindrops are mostly round when sunlight refracts through them, the visual effect is a spherical arc that soars above the sky.

These refraction and reflection process described above creates a primary and secondary rainbow. The first and brighter is called the primary rainbow. The second and more faint rainbow is the secondary rainbow, it occurs when the refracted light does not escaped the raindrop after being reflected the first time. Instead, the refracted light reflects off the raindrop's surface a second time, producing a secondary rainbow with its color reversed compared to the primary rainbow. There are only fewer light ray available to undergo the additional refraction process, so the resulting secondary rainbow appears  less vivid.