This is a good biological explanation. The physical explanation is, if the sensitivities didn't overlap, our spectral sensitivity would not be continuous. There would be valleys of zero sensitivity between the cones, and a continuous wavelength sweep would result in us seeing black bands between colors.
Gray bands, or more realistically just desaturated bands. There'd still be sensitivity to light through rods (black and white), and even if the peaks of wavelength sensitivity were highly separated there would still be some cone response to wavelengths that didn't stimulate them strongly.
