The best answer is a mix of a couple things. In order to imagine how we would see a new color, instead we need to imagine what it would be like to see one less color.

Right now in the colors of subtraction we have yellow red blue and the colors of addition we have green red blue (if i got those reversed, you can correct me.)

I'd love it if you could download a program on your phone called twilight, which removes blue light from your phone at night. If you get used to it, you almost never know when it's on. This makes you believe that without the color blue, we probably would never know, everything is just different shades of red and green.

Certain animals cannot see the same colors we can, so if we presented let's say purple to a dog that couldn't see blue, yet could see red, they would be able to tell it is different from red, however they would percieve it mixed with another color they could see, like yellow for example.

Red green colorblind individuals usually say they see shades of orange/yellow instead of red/green. This varies from person to person.

Now that we know not being able to see a color that others can doesn't magically create a new image in our minds, instead our minds compensate using colors we can perceive. We know a bit of what it would be like if there were a fourth color in our spectrum, but it would be impossible to imagine even if you were looking at it.

I guess in theory if we find a new type of light, we might have the opportunity to see new colors.

One interesting fact, somewhat related is a phenomenon of cataract surgery patients being able to see more vivid colors after, because one treatment removes an amber "filter"(for the lack of a better term) from your eyes that everyone is born with.