What is the area between the two rainbows called, and why is it darker than the rest of the sky?

The area between the two rainbows is called Alexander's Dark Band, after Alexander of Aphrodisias, not after Alexander's Ragtime Band, which is an entirely different thing.

To understand Alexander's Dark Band, you need to understand the rainbow. The bottom rainbow is the primary, and is brighter. It is formed as sunlight enters the raindrops, is refracted (bent), reflected from the back of the drops, and refracted again as it leaves the drop. This separates the light into the spectrum of colors, and also scatters some of it downwards, toward the inside of the rainbow, making that area brighter.

The top rainbow, called the secondary rainbow, is formed by sunlight reflecting twice inside the raindrops, resulting in a reversed rainbow. Instead of the red band being on the top, as in the primary, the red band is on the bottom in the secondary. The scattered light from this rainbow is scattered towards the outside of the rainbow, making it brighter. The Dark Band looks dark because the rest of the sky around it is made brighter by the rainbows.