It is understandable that life didn't evolve the capability to sense all wavelengths of light, it simply didn't need it. My question is, why did it evolve the set of wavelengths it has, for example, the human visible range is 390-700nm, so why couldn't it be a different set of wavelengths, like, (say) 300-620nm? Is there a reason for it?
Well humans actually have a reduced spectrum compared to non-mammals. Reptiles and birds have 4 color sensitive cell types (cones), mammals lost two of these as modern mammals are descended from early mammals who were nocturnal. Primates evolved a third cone, (a mutant variant of one of the two they had before) Primates did this because many a frugivores and color is excellent for determining when fruit is ripe.