Sensations and physical quantities
Color is a sensation: it exists in the mind; it is not a physical quantity, that is, it does not exist in nature. Since color is a sensation, all measurements of color are made with reference to a hypothetical "standard observer"; the color discrimination ability of most men with normal color vision is somewhat poorer than standard observer's, that of most women is somewhat better. Importantly, what is measured by a color measurement does not necessarily match what is perceived by the brain -- the measurement is related to the sensation of color only for extended objects, that is, objects which subtend a large part of the visual field; for smaller objects other mental mechanisms come into play which make a mess of the "objective" measurement. For illuminating examples, see professor Akiyoshi Kitaoka's illusion pages, for example, page 13.
For extended objects (objects which occupy a large part of the visual field), the sensation of color is related to the physical quantity spectral density of light; the relationship between the spectral density of the light and the perceived color is complicated but predictable using empirically measured formulas. For smaller objects the perceived color cannot be determined from the power spectrum of the light coming from those objects; only by taking into account the entire scene can the color be predicted (approximately), and there are no good formulas.
RGB is not enough and cannot be
The International Commission on Illumination (CIE, Commission internationale de l'éclairage) carried out an extensive set of experiments which determined that the standard observer can match any given color by varying three parameters. (Essentially, the test subjects had to match the color of a light source with the color of another light source which could be modified by turning three knobs.) Mathematical calculations showed that combining three abstract light sources is enough to match any visible color; unfortunately, the three abstract base colors which define the CIE 1931 XYZ color space are non-physical, that is, they cannot exist. (They are a red much redder than the reddest visible red, a blue very much bluer than the bluest visible blue and a green a little greener than the greenest visible green.)
In practice, we either accept than any three visible fundamental colors will be able to reproduce only a part of the visible colors, or, if we truly want to reproduce a larger part of the visible color, we accept that we need more than three fundamental colors. For example, high-quality color reproduction on paper is done in hexachromatic processes; for color reproduction on screen advanced televisions use four base colors.
In particular, the commonly used sRGB color space can reproduce less than half of the visible colors, which is understandable given that its primary colors are chosen so that they match the colors of the phosphors available for color TV screens in the 1950s. In particular, the green primary of sRGB is very poor; sRGB simply cannot reproduce luminous saturated greens.
CIE 1931 xy chromaticity diagram showing the gamut of the sRGB color space and location of the primaries. By Spigget, made available of Wikipedia under CC BY-SA 3.0.
(There exist computer monitors and television sets which can reproduce a wider gamut than sRGB; but the problem is that (1) they are very expensive and (2) the overwhelming majority of visual media are encoded in sRGB. Look for "wide gamut" monitors, which are not the same as "deep color" monitors. Even monitors which can faithfully reproduce more than 95% of the sRGB gamut are quite expensive.)
So how come we can use sRGB?
Remember that color is not a physical quantity, but a sensation which exists in the mind. The perception of color is seldom absolute; in most practical situations it is the color contrast which counts. As a consequence, cameras and image manipulation software which use the sRGB color space cheat by mapping the visible colors to the smaller representable gamut. As a commonly encountered representation in the user interface, you may seen some color printer drivers offer a choice of intents when reproducing colors; common choices include pictures (mapping all colors to the device gamut) and presentations (clipping colors to the device gamut).
How would an alien perceive our screens?
All the preceding discussion intended to convey that the perception of color is different for different humans, no aliens needed. But to come back to the question: we are obviously unable to picture the sensations of an alien which has a color vision system with more primary stimuli than ours; but we can make an attempt at an analogy.
All user interface designers are taught that colorblindness is a thing, and that about 10% of human males have less-than-standard color vision; for this reason, user interfaces should always be checked for usability by colorblind people, and there are countless software programs which attempt to simulate what a colorblind person sees.
A beach scene in natural colors and simulated red-green color blindness. Own work by AlexP, simulated color blindness made using the Color blindness simulator from Etre.
We can intuitively imagine that for the multi-chromat alien our screens will have a similar relationship to a full-alien-color image as the simulated colorblindness images have to our full-human-color pictures. This is a gross simplification, obviously. One immediate example of why this is an oversimplification: imagine that the alien sees yellow as a primary color, whereas humans cannot -- we cannot distinguish between a mixture of red and green and monochromatic yellow; to the alien, all yellows in the picture will appear as weird color shifts.
But would they be able to read the screen?
Well, that really depends on what part of the electromagnetic spectrum they perceive as visible light. If their visible light overlaps with ours then yes, they will be able to read black text on white (or white text on black), although for them the white may not be white but some color. They may even be able to read some color combinations which are hostile to a human reader, such as the infamous magenta text on green background...