Here's the deal: Color comes in three Primary colors: red, green and blue.
You can generate any color by (starting with no color, black, and) adding these in some combination (additive color). Doing this requires three color sources, each providing one of the primary colors. If you look closely at your (color) TV, you'll see the image is comprised of tiny red, green and blue dots.
Here's how they combine to create Secondary colors:
It really does work that way; I've done it.
In a dark theatre, I took three lights and used filters to make them red, green and blue. I pointed them at a white screen in a pattern much like the above. And much like the above was what I saw! It was pretty amazing to see white light where you'd think there'd be none.
When you're talking about paint or ink–any reflective color media–you're talking subtractive color. The key here is that reflective color starts with white and removes primary colors. By combining two Secondary colors you remove two unshared primary colors, leaving the common primary.
Here's how they combine to create Primary colors:
People argue about this one a bit. It has a little to do with how some people name colors; some conflate magenta and red; some conflate blue and cyan. And it makes some difference if you're dealing with paint, transparent ink or colored filters.