Why does the human eye only sense the visible spectrum of light

Please provide a clear explanation

The reason is because of our sun!

The sun emits light at a wide range of wavelengths but, if you look at the spectrum, the peak intensity occurs around about 500 nm: http://en.wikipedia.org/wiki/File:Solar_…

Therefore, in evolutionary terms, it makes sense that our eyes should be very sensitive at about 500 nm because this is where we get maximum illumination of our environment from the sun. And, unremarkably enough, this is the wavelength at which the rods in our eyes are maximally sensitive (the rods give us intensity information but not colour).

So the reason our eyes are sensitive to the 'visible' part of the spectrum is simply because these are the most intense regions of the sun's output and most of the objects in our environment are made of matter that reflects and absorbs in these wavelengths, giving us subtle shades we can use to identify objects.

well, the facile explanation would be that what the eye sees is the very definition of visible
but
let's go a bit deeper

light is a part of the electromagnetic spectrum
it is not fundamentally different from radio waves
it differs only by frequencies and wavelengths

radios pick up certain frequencies by virtue of the size and configuration of their antennas
(they further narrow the frequencies by using a tuned circuit)
anyway
AM radios pick up a certain range of frequencies (from about 0.5 MHz to 1.6 MHz)
FM radios pick up from about 85 MHz to 110 MHz

our eyes have rods and cones which respond to certain frequencies due to their size, shape and chemical composition
these receptors are able to respond to EM waves from about 400 THz (400E12 Hz) to 800 THz
so
this rather small range of frequencies are 'visible' to the human eye

rods are fairly broad spectrum receptors
mostly rods respond to light and dark and, therefore, motion