Why do all national flags, bar two, have either red, white, or blue? And many flags having all three? Why are these three colours so incredibly popular?

I can't explain red and blue, but as to white...

In heraldry, the colors are:

Gules (red)
Azure (blue)
Vert (green)
Purpure (purple)
Sable (black)

The metals are:

Or (gold or yellow)
Argent (silver or white)

The rules of tincture in heraldry do not allow you to put a color on color or metal on metal. This is actually a totally common sense rule, because if you break it you'll have terrible contrast--and you want good contrast to be able to identify a flag from a distance.

(There are some exceptions, and it gets a little more complex. If a charge's tincture is "proper" -- meaning "true to life" -- it might be allowed to sit on a color or metal. And if the field (background) has a division of color and metal or a color, then a color or metal charge can go on top, such as in the Algerian flag.)

Consequently, most flags have either white or yellow. You might have also noticed school and team colors almost always have white or yellow.