This question is related to the question: Why are some things transparent and others opaque?
Being able to see something requires that it is opaque and that sufficient light illuminates it.
UV and shorter wavelengths are not as prevalent as visible light on earth. The world would appear too dark to see if we used UV and shorter wavelengths. This is because our atmosphere absorbs most high energy light.
Infrared and longer wavelengths of light pass through many objects which would make vision difficult. There's less light here reaching the earth and even less being refracted.
Think about how much our vision relies on indirect light. The frequencies at which most objects are opaque makes those frequencies useful for vision due to the accumulation of refracted light.
Why are many objects opaque in the visible spectrum of light? Longer wavelengths of light have less energy than the valence electrons on most matter. Shorter wavelengths have too much energy, they cause chemical reactions, in addition to not being very prevalent on the surface of the earth.
Electrons are what absorb then remit light and have thresholds based on their chemistry for what they can absorb. No absorption = transparent. Too much energy and chemical reactions start to happen, which may be undesirable, or desirable in the synthesis of vitamin D by UV light.
Plants extract energy for chemical reactions from wavelengths shorter than infrared, which is too weak to drive photosynthesis, and not as abundant as visible light. But also absorb wavelengths longer than ionizing frequencies, which are not very prevalent, and usually cause damage.
Visible light is the spectrum of light which is prevalent enough on earth to see, but is not so energetic it would harm biological systems. The qualities for optimal light frequencies in sight and photosynthesis overlap because they have similar mechanisms for interacting with light. What is this mechanism? The chemistry of carbon based life.