Why is it that colours of the visible spectrum (e.g. orange) can be "faked" by mixing other colours (e.g. red and yellow)?
The retinas in your eyes contain three types of color-sensitive cells called cones. Although each type of cone is sensitive to a broad range of wavelengths, the blue cone has peak sensitivity to blue light, the green cone is most sensitive to green light, and the final type of cone is most sensitive to red light. All colors that you see are really your brain's interpretation of the combined signals from your cones. For example, your green and red cones are about equally sensitive to monochromatic yellow light from the visible spectrum, so if your brain gets equally strong signals from your green and red cones, it will label a color "yellow." If you shine two equally strong red and green lights onto a white screen, you will see yellow! It doesn't matter whether the light source is truly yellow, with a wavelength of about 590 nanometers, or whether it's a mixture of red and green light--your brain will interpret the color as yellow regardless.
Computer monitors and TVs are designed to mimic the eye's cones and have miniature red, green, and blue light sources. By varying the brightnesses of the tiny light sources, the monitor can mimic almost any color a human can see. Having a different number of cone types would change both the wavelength range you can see and the way mixed light colors look to your eye. For example, cats and humans who are red-green colorblind lack the red cone and so confuse reds, oranges, yellows, greens, and browns. See this page for an example of cat color vision (scroll to the bottom):
Why does the colour purple (i.e. the mixture of red and blue) look similar to or exactly like violet (i.e. the visible colour of highest frequency)?
The L-cone (red) and M-cone (green) are somewhat sensitive to violet light. See the linked graph of cone spectral sensitivity.
Monochromatic violet light, from the visible spectrum, triggers a strong signal from the S (blue) cone plus weaker signals from the L and M cones. Therefore, if your eyes see a combination of lights that strongly trigger the S cone and give weaker signals to the L and M cones, as you would get from mixing blue and red, you perceive violet.