Well first, take a look at a spectrum (like a rainbow) of the light to grasp that white light includes a multitude of colors.
But WHY does mixing, say, as Matt King says, red and green PAINT offer an entirely different color than mixing red and green LIGHT?
Because the paints filter light; that is, they remove light. Although it's mixing, it's quite close to putting translucent layers (think of wratten filters) on top of each other. White light from the sun or artificial lamp shines through the red paint layer. The red paint layer lets red light go through, destroying every other color. Then, the resulting red light goes through the green paint. Because green paint would let only green light pass, and there is no green light left to pass, the result is black - all light has been destroyed, partly in red pigments, partly in green pigments. In practice, the paints are far from "perfect" red and "perfect" green, thus you don't get black but some kind of ugly brown; not all light is absorbed by these paints.
So, pure red paint could be called "remove everything from blue to yellow" paint.
By mixing more paints, you remove more colors.
This is simply because no pigment can create light. They don't shine in the dark. The only way they can work is by removing "unwanted" parts of spectrum.
If you mix more paints, you remove more colors. Pure, vivid red and green paints BOTH remove yellow, so you make double-sure you won't have yellow when you mix them.
So, paints that remove large sections of spectrum are not used that often: pure red, green or blue let only a small fraction of light pass, removing almost all other colors. So you don't need to mix them with anything else to remove anything more.
Instead, paints with broader spectral passband, for example, yellow are used. Yellow paint (usually) removes only blue, and lets everything from green through green-yellow, yellow, orange to red pass through. This creates you an opportunity to FURTHER remove colors by mixing in another paint.
As for your original question, it's only a terminology play. "Primary colors" are just some arbitrary colors that can be selected. Then, in that same system of your choice, the "secondary" colors are colors created by mixing any two of your "primary" colors.
There are two widely used systems: additive where you create light, and subtractive where you remove light. In the additive system, primary colors are RGB. In the subtractive system, primary colors are CMY.
They teach that painters use red, yellow and blue. This itself is a big lie. Well, they look a little bit like red, yellow and blue, but the red and blue are FAR from vivid red and blue. Especially the blue is much more like cyan. The "traditional" painting colors thus are somewhat close to CMY but not quite, so they just cannot produce pure colors. If they were really red, yellow and blue, you couldn't mix them practically at all.