The human eye, or more specifically the retina, has three types of "cones"or "light receptors" that are sensitive to different frequencies of electromagnetic radiation (light). As it turns out, the cones are sensitive to wavelengths (or frequencies) around red, green, and blue light. Incoming light (at visible wavelengths) will generally stimulate all three types of receptors to different degrees, and the brain will combine this information into color and intensity.

We cannot see UV light because the light receptors in our eyes are not sufficiently stimulated by UV light. Part of the "problem" is that the cornea and your eye lens absorb much of the UV light. In addition, I think because of the lens shape, the UV light is not focused on the retina (which would make things blurry), but you might want to research this to double check me.

Also, I'm not too familiar with how sensitive the photoreceptors in our eyes are to UV light (if enough of it got through), so you might want to investigate that as well. I think some insects see in the UV (honeybees, for example), as well as some birds andreptiles. There are probably others as well.