Across the electromagnetic spectrum, 400-700 nm is a narrow spectrum of frequencies and focused in the region of short wavelengths. For example, radio waves cover a large range of frequencies unexploited by the visual system. So what biological reason is there that evolved us to use such a small frequency bandwidth for vision?
The visible spectrum has the highest energy in sunlight at the earth's surface, explaining the gross location of the visible spectrum in life on earth. The specific frequency range varies across species and can be explained by species-specific survival strategies.
When you look at the solar light spectrum at the earth's surface the visible spectrum has the highest intensity (fig. 1).
Solar irradiation. Source: University of California.
So it makes sense to use the range of frequencies that is represented most in sunlight as a starting point.
Then the question becomes, why do humans utilize approximately 400 to 700 nm, and not infrared or UV? That can be explained because we do not need it. Our range has been hypothesized to be related to foraging behaviors and our visual system is particularly sensitive in the frequency range of the coloring of (ripe) fruits, which is thought to have been of great benefit to our hominid ancestors (Osorio & Vorobyev, 1996).
Why then do animals extend their vision into UV? Many fish, amphibian, reptilian, avian, and some mammalian species use UV vision. Many birds can identify UV-reflected nectar and berries, and UV-reflecting plumages in birds, and scales in fishes are used for recognition (Shi & Yokoyama, 2003). Moreover, some arthropod species are know to use UV vision to reduce light-reflection distortions under water, such as in the mantis shrimp that features 12 photoreceptor types (as opposed to four in humans) (Thoen et al., 2014).
Why then do animals extend their dynamic range into the infrared? A notable beneficial effect of perceiving infrared is the detection of body heat. The generation of heat is accompanied by the generation of infrared light. The detection of this emitted light is highly useful for nocturnal predators, like the rattle snake (Hartline & Newman, 1982).
- Hartline & Newman, Sci Am (1982); 246(3): 116-27
- Osorio & Vorobyev, Proc Roc Soc B (1996); 263(1370)
- Shi & Yokoyama, PNAS (2003); 100(142003): 8308-13
- Thoen et al., Science (2014); 343(6169): 411-3
Most of the light from the sun doesn't actually reach the earth's surface due to the atmosphere.
So the light reaching earth includes near-UV, visible, near-IR and a band of radio waves. Seeing any other part of the spectrum would be impossible since it doesn't reach earth.
You asked why we only see in the visible light range; this is due to evolution. Birds, among other animals, can see UV light. In fact, all vertebrates have the potential for near-UV vision. Humans, as vertebrates, also have UV-sensitive photoreceptors. However, our lens is opaque to UV light:
[from Clinical Ocular Anatomy and Physiology via this website]
Just speculating, it would seem that the ability to see near-UV is an ancestral condition which we lost at some point; either it provided no significant advantage (neutral regression) or there was some advantage provided by a UV-opaque lens, either directly (such as protection from UVB light) or indirectly (through pleiotropic antagonism).
On the other hand, human photoreceptors cannot detect IR light. Again, this is a product of evolution. These researchers hypothesize that the longer the wavelength of light detected, the more noise is produced. This noise is due to activation of the pigment molecule by heat. Or, it could just be that never happened. An IR-sensitive photoreceptor might be possible, but evolution doesn't lead to perfect adaptation. In other words, there isn't necessarily a reason why.
As for radio waves, they are too low energy to interact appreciably with matter, at least as far as vision is concerned.
Its not really
That is only the human visible spectrum.
Humans actually have a reduced spectrum compared to many animals. Mammals in particular have a reduced spectrum compared ot non-mammals. Reptiles and birds have 4 color sensitive cell types (cones) and can see into the ultraviolet. Many invertebrates can see an even wider spectrum. Mammals lost two of these cells. Modern mammals are descended from early mammals who were nocturnal Thus color vision was less useful. Primates evolved a third cone, (a mutant variant of one of the two they had before) Primates did this because many are frugivores and color is excellent for determining when fruit is ripe.
This is my speculation, but there are no or few organic chemicals which can absorb radio waves having longer wave lengths. Eyes sense lights by absorbing light with organic chemicals, but to sense longer wave length, eyes might need more sophisticated devices like tuners.
In addition, less than 100nm radio waves from the outer space are absorbed by the atmosphere, so even if eyes could sense such radio waves, you don't see any signals on the earth.
"Visible light" is the wavelength of the light that we can see, if human beings could see UV or IR, those wavelengths had been included in the "visible light" definition. Now, our Sun is brightest in yellow-green light, which (you guess?) is right in the midle of the "visible light" spectrum. So the only remaining question is: "Why can we only see that short spectrum?" The answer is EVOLUTION. The same reason that made us to have only 5 fingers in our hands, only 2 eyes and only two kidneys. The bandwith of the "visible spectrum" is enough to make the human beings (as specie) survive.
These wavelengths happen to satisfy two conditions.
First, shorter wavelength photons (EM radiation) tend to be dangerous for biology. Even UV light (<350nm) is already can damage DNA. That is due to the fact that bond energies in biological molecules tend to have values close to energy of short-wavelength EM radiation. These is why high-energy particles called ionizing radiation. For reference, see this paper. ionization energy for DNA estimated in 4-5eV range, which is 300-250nm.
Secondly, on low-energy hand of spectrum, water absorbs a lot of IR radiation:
So, as you can see, what we see aka visible spectrum is sitting nicely in valley of water absorption but does not extend to area of harsh UV light that will damage chemicals of your body.
Now, why is that so? Because atoms have these masses and electrons around them have these energies. So bonds and molecular interactions have these values.
(In radio range of frequencies you, opposite to UV, have not enough energy to elicit any significant molecular changes, which is why you can't perceive WiFi)