I'm only just recently getting into colour theory, and so far I have found this to be the most shocking and confusing part.
Assuming you are familiar with the concepts of complementary contrast and saturation in broad lines (if not only through the answer of David D), the following anecdote of Goethe's 'discovery' of complementary contrast is very informative in regard to your question. His eyes were once caught by a dark-haired girl in a pub, and, when she moved away, and Goethe was left staring at a white wall, he saw that same girl with dark skin and light hair - the negative of the real image. *
According to Johannes Itten
two colors are complementary if their pigments, mixed together produce a neutral gray-black.
Physically, light of two complementary colors, mixed together, will yield white. Two such colors are a strange pair. They are opposite, they require each other. They incite each other to maximum vividness when adjacent; and they annihilate each other, to gray-black, when mixed - like fire and water.
There is always but one color complementary to a given color. In the color circle, complementary colors are diametrically opposite each other.✝
In the image below, the colours on both sides of each strip are opposites, and it is shown that mixing them more and more with the other colour desaturates them, and finally yields a neutral grey colour in the middle:
The colours in a complementary contrast are each other's opposites, they cancel each other out. Because of this, they not only desaturate or attenuate the colour intensity, but also the temperature, the value (or lightness), and hue.
So, practically, you can create greys in paint by mixing complementary colours, or get a more neutral colour when working with lighting when combining complementary coloured filters. This works both in additive and subtractive colour mixing.
Physically, the complementary colour of any given colour can be found by mixing all other colours: "for every hue, the sum of all the other colours in the spectrum is the complementary of that hue". ❧
Just to get some terms defined:
Color saturation refers to the intensity of color. When color is fully saturated, the color is considered in purest (truest) version. Techopedia.com
a color that combined with a given color makes white or black OED
In painting, here's why a color's complementary color reduces intensity (desaturate)
First, each of the six pairs of Complementary Colors always contain one cool and one warm Hue. This means they are the colors with the most contrast.
Each pair of Complementary Colors always contain some combination of all three Primary Colors. For example, in the pair Yellow and Purple, Yellow is one Primary and Purple is a mixture of the other two Primary colors Red and Blue.
When you mix all three Primary Paint Colors together you get a dark neutral. So because Complementary Colors always contain all three Primary colors, they quickly neutralize each other when mixed together.
From Color-Wheel-Artist.com There are some very good illustrations at the link.
Bottom line: When you mix complementary colors you are blending all three primary colors which reduces intensity/saturation.
I thought I'd attempt a more scientific answer:
A mix of two complimentary colors is less saturated, because complimentary colors are composed of wavelengths that trigger different cones in our eyes. When combined, all cones are triggered more evenly, which makes our brain perceive the result as less saturated.
First, I think it's important to keep in mind that "color" is a human construct to describe things we see - it makes color theory easier to understand. Colors are not consistent across languages (although there is some logic to them), some "colors" (e.g. silver vs blue) describe different optical qualities, and so on.
So, what are colors? Everything we see is because of light. Light comes into our eyes and hits photosensitive cells, and some of these cells are sensitive to different wavelengths. Most people have three types (some have less, some have more). Our brains perceive these three signals as colors: blue (shortest), green (middle) and red (longest). (more info)
When all types of cones are stimulated evenly, we see white (or grey, depending on the amount of light).
"saturation" is one of the terms we've created for describing colors. Saturation describes how "bright" a color is, or from the other end, how far from white it is. Therefore, we can roughly say that a saturated color hits only one or two types of cone, while desaturated hits all types to some degree.
The color wheel
So, how does one take wavelengths and eye cones and get a circle? The color wheel is a tool created by painters to predict how will different colors behave when mixed together. We figured the wave stuff after we made the wheel. The fact that the new science didn't break it is a testament to artists being really damn good at observing and describing cause-effect relationships.
So, in a way, we can say that we call colors "complimentary" because they desaturate each other. They're on the opposite side of the color wheel, because that's what makes the wheel work as a predictor.
So, why do complimentary colors compliment each other? It's because they consist of wavelengths that trigger the other cone types. If one color triggers one cone type, the other triggers the other two. When you put complimentary colors together, the cone types are triggered more evenly, which makes them look closer to white, therefore less saturated.