Summary: human vision has four physiological components giving rise to three color components in opponent color theory. The CIE chromaticity diagram was constructed by realizing all colors can be created by varying intensity of two appropriate monochromatic sources. Reaching a color by two different combinations is meramerism, and the colors are metamers, and indistinguishable. Computer displays have three monochromatic components interfering with a backlight. The nature of component masking reducing intensity with three components results in a triangular gamut strictly inside the CIE chromaticity diagram, enclosing a white point. All of the colors in the gamut are reachable by continuous variation of monochromatic intensity. The gamut has straight sides by definition of the CIE chromaticity diagram.
Typical human vision involves four components: one type of rod which excels at accumulating the effects of sequentially arriving photons, and three types of cones: short, medium, and long. All four are sensitive to light across a range of wavelengths in the general shape of a bell curve. However, peak wavelength sensitivity differs among the four. Short cones are most sensitive to violet wavelengths, medium cones to green, long cones to yellow, and rods to blue-green. The following image shows the relative response profules for each, with cones color coded. The black line profile is for rods.
A theory of human vision is the trichromatic model. This model of human vision, which leads to the CIE chromaticity diagram, is essentially that the rods are responsible for controlling the overall perceived intensity, while each cone is responsible for varying the position in the chromaticity diagram. One can determine the position by integrating the input wavelength profile against the three bell-curve-like responses of the cones. Repeat the integral for the rods and multiply that by each of the cone integrals to obtain a vector with three components. The result is a position in the overall gamut of human vision. Note that the chromaticity diagram is two dimensional, while the vector is three dimensional. The chromaticity diagram is a slice through the gamut of human vision, which has the shape of a cone in the trichromatic model. When intensity is very low, rods and cones receive no photons, so all colors tend to lose saturation and become what we call black, reflected by low values of the rod integral.
The CIE chromaticity diagram was constructed by experiment. Participants were asked to match a displayed color by adjusting the intensities of various light sources. The adjustable light sources were comprised of a variety of near-monochromatic profiles. The lights were controlled by knobs whose positions (and thus intensities) could be recorded. It turns out different participants would match the colors using different combinations. The same color produced by different combinations of light are called metamers. The diagram was constructed by realizing that interpolation of relative variation of different monochromatic sources led to some of the same colors. Put another way, there was overlap if different monochromatic sources were used. To sum up, if you can reach any two positions on the chromaticity diagram, you can reach any position on a straight line between them by varying the relative intensities of your sources. If you have three non-collinear sources, you have a triangle, and can reach every point in the triangle by the same method of varying intensities.
Shifting gears, consider the properties of consumer computer displays. They have some form of color control, almost always with the three components red, green, and blue. The control generally involves a mask which can be varied continuously for each component at each pixel. They also have some form of backlighting which, by design, is as close to standard white points as is commercially feasible (D50 or D65, depending on requirements). Maximum intensity is only reached when all three components are fully engaged. When any component is disengaged, even partly, the overall intensity decreases. When only one component is engaged, much of the backlight is blocked, intensity is reduced, and the extremes of the chromaticity diagram can't be reached. So any display gamut will be strictly within the bounds of the chromaticity diagram. We also only have three monochromatic components to work with, so we expect the gamut of the display to be a triangle, by definition of the chromaticity diagram. Furthermore, every point in the triangular gamut is reachable by varying the intensities. Finally, because the backlight is a standard white point, we expect white to be somewhere inside the triangle.
Hopefully this answers your questions!