Iridescent clouds aren't too rare, but they are very pretty. And these particular clouds let you see exactly how they work, by showing where the sun is, and what clouds do and do not turn into rainbows.
Iridescence in rainbow clouds happen the same way iridescence in soap bubbles does. Light runs into very, very small objects, and is bent around them. This often results in two different waves of light interfering with each other. In most cases, the light would interfere as a whole and result in patches of light and dark. But in the case of soap bubbles and iridescent clouds, the different wavelengths of light are separated out. So in some areas, the color red destructively interferes with itself - or with other wavelengths - and drops out of the light entirely, leaving it to look blue-green. In other places, red constructively interferes with other wavelengths, emphasizing the red light and making the area look bright red or pink.
It seems the more light there is, the more you would see these clouds. Partly-cloudy days should be full of rainbows. But too much light does its part in obliterating these clouds. As you can see in the pictures, the areas where the sun is brightest wash out the rainbow. It's possible to see your reflection in a darkened window, but turn on the light and the weak reflection is drowned out with the brightness. On bright days the slight edge of iridescence that some parts of the cloud gain gets drowned out out when compared to the rest of the scattered light from the cloud.
This is why you generally see iridescent clouds in the evening or the morning, or when some structure obscures much of the light from the sun. You can also generally see more iridescence when you're wearing sunglasses. The size of the droplets has to be just right, and there has to be so little of it that the overall light source doesn't erase the rainbow. So, if the end of the world comes a little late and with low light, do try and look up at the sky.
Image: The Brocken In a Glory