Mice are colorblind, so they see colors similar to the way red-green color-blind people do. That doesn't mean they don't see any colors at all, but they can't see many. They look at the world in shades of gray and a few additional hues like dull yellow and blue.
Why You're Seeing Red
Mice have two cone pigments in their eyes -- giving them dichromatic vision -- while humans have three cones, for vision that's called trichromatic. Each cone contains a different photopigment that's sensitive to a different wavelength of light -- either blue, green or red. When combined, these wavelengths show us the full spectrum of colors. So having that third cone allows us to see red and distinguish the difference between greens, reds and yellows -- something colorblind people and most mammals can't do. Their vision is dichromatic.
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The only other mammals with three cones are Old World primates like macaques and baboons. New World monkeys, like squirrel monkeys, have an intermediate color vision that's partway between those of humans and other mammals, and some female New World monkeys have trichromatic vision.
Mouse Experiments
Scientists have helped some mice to see a rainbow of colors. A study published in the journal Science in 2007 details how lab mice were developed with three cone types. Gerald Jacobs of the University of California in Santa Barbara, and his colleague, genetically added a piece of DNA into mice's genome to produce the red retinal pigment. The experiments allowed the mice to see in full trichromatic vision rather than their normal dichromatic view of the world.