The visible part of the spectrum is associated with (relatively) low energy electronic transitions in molecules. The photoreceptors in your eyes operate on the basis of cis/trans isomerisation of a carotenoid molecule confusingly called retinal, where a double bond is temporarily turned into a single bond by a photon, causing part of the molecule to flip. If my memory serves me correctly, the excitation that gives rise to this isomerisation is a $\pi\rightarrow\pi^*$ transition associated with the delocalised electron cloud that results from the conjugated electronic structure of the molecule (that is, those alternating double and single bonds). Conjugated structures typically absorb strongly in the near UV and visible parts of the spectrum and are brightly coloured due to their absorption or fluorescence spectra. Metal complexes are also typically brightly coloured due to excitations involving electrons moving between the metal atom and the ligands connected to it. The vibrant red of blood and (somebody please correct me if I'm wrong) the green of chlorophyll are attributable to charge transfer bands - in the case of chlorophyll the strong absorption this molecule (actually, a family of molecules) has in the red region is pivotal to photosynthetic light harvesting. Many other pigments (carotenoids, melanin) in both plant and animal systems are involved in photoprotection - sacrifically soaking up high energy photons.
But I digress - if we go further up in energy (shorter wavelengths, into the UV region), the photons become energetic enough to liberate electrons from molecules and materials entirely. The photoelectric effect is a manifestation of this in metals. As such, ultraviolet light is considered ionising radiation and has a habit of wrecking molecules. When you go beyond this into the X-ray spectrum, you can actually knock out the most tightly bound core electrons of an atom, with disastrous effects. Gamma rays are even more energetic, and can interact with the nuclei of atoms.
If we go lower in energy (longer wavelengths) we end up in the infrared and microwave regions of the spectrum. These photons lose the ability to directly excite localised electrons (though microwaves can interact in a special way with the delocalised electrons of metals, which is why you shouldn't put metals in the microwave. Instead, these lower frequencies tend to contribute to the vibrational and rotational motion of molecules.
Now, many animals are sensitive to frequencies outside of the human visible range, in both the UV and IR regions, however the visible range, associated with fairly gentle electronic transitions, is something of a sweet spot.