Before getting into scientific reasons, there is an issue of definition.
By definition, invisible EM waves cannot be seen.
That aside, there are two main reasons that invisible EMs are invisible:
1) the photoelectric effect:
https://en.wikipedia.org/wiki/Photoelectric_effect
Longer EM waves cannot trigger the electron changes in the rods and cone of our retina.
They simply do not pack the energy to do this.
2) the transparency of our eye lens and the liquid in our eyes:
In general, transparent materials are transparent in the visible spectrum - with no guarantees for other wavelength. In particular, the water in out eyes is very transparent to visible light, but will block most microwave radiation.
3) focus:
Different wavelengths are focused differently by the materials in our irises. Those materials are "designed" for visible light. While looking at the violet light coming from black-light bulbs or bug zappers, you may have noticed that the bulbs look blurry. This is because when you get to the edge of the visible light spectrum, especially the high end (UV), our eyes do not focus these wavelength onto the retina very well.
There is another way of interpreting your question. Since the eyes work as "designed", why haven't they adapted to other wavelengths.
That answer is:
* On the surface of planet Earth, day light includes lots of visible light.
* Visible light is on the border of "ionizing radiation", so an eye that focused UV onto its retina would then need to accommodate the destructive aspects of UV radiation on proteins.
* In the kind of scenes that mammals developed in (last 100M years), near IR does provide some additional information not available in visible light. For example, if you cut a plant from its roots, it will lighten in the near IR long before there is any visible change in the leaves. Apparently for humans and our predecessors, this additional information was not enough to develop another set of cone cells on the retina. But, compared to most other animals, we do really well with our color vision. The point is that there is a trade-off between gaining additional useful sensory information and the "use" of evolutionary development.