You should study more about how the eye works
Color is the result of your brain interpreting signals from the eye. The human eye has photoreceptors that are sensitive to the certain wavelengths of light that we call red, green, and blue. Take a look at this chart from the Wikipedia article on color - it shows how much the different photoreceptors react to a given wavelength of light. As others have mentioned, all colors we see are simply all the various combinations of stimulation levels of our red, green, and blue color photoreceptors.
Consider the color yellow. If you are exposed to light with a wavelength 575–585 nm, your eyes will send a signal to your brain that gets interpreted as seeing something yellow. If you look at the chart again, you'll see that at 575 nm, both the red and green photoreceptors should have a fairly strong reaction. Computers take advantage of this when displaying colors to you - the RGB value for yellow is #FFFF00. For anyone unfamiliar with RGB values, this is interpreted as 255 red, 255 green, 0 blue. So a computer doesn't produce 575 nm light, it gives you a mix of red and green light. The end result is that your red and green photoreceptors both react - the exact same signal for 575 nm light.
So how could someone see new colors? Their brain needs to receive more information. Suppose you had a fourth type of color photoreceptor in your eye sensitive to light between green and blue. Without that fourth kind, a mixture of green and blue (#00FFFF) is interpreted as cyan. With the cyan photoreceptor, you would be able to see more colors - a mixture of green and blue light would strongly stimulate those two photoreceptors and only weakly stimulate the cyan one, while cyan light would strongly stimulate the cyan photoreceptor and weakly stimulate the green and blue ones. This means that your brain would have a way to distinguish between cyan and a mixture of green and blue, and as a result you would see those two scenarios as two different colors.
Could a colorblind being use something to see more colors than us?
A completely colorblind species would receive a relatively simple signal from their eyes - how much light is coming in. To see more colors than us, they would need to receive more information than us. Their eyes are not suited for this task. Special goggles can't fix this.
If you're thinking about the glasses that humans with certain kinds of colorblindness can use, that's a fundamentally different situation. The short explanation is that those people still have three kinds of color photoreceptors but they overlap even more than normal. The glasses are able to correct for that extra overlap. Again, their eyes are still sending three color signals to the brain.
So in order for the "special goggles" you mentioned to work, they have to bypass the alien's eyes. However they would do it, they'll just send more information to the alien's brain than our eyes do to our brains.
Interpreting that extra information is a separate problem. You could solve it by giving the aliens synesthesia as @Willk suggested - the color information comes through one of their other senses.
You could also solve it via neuroplasticity. When they first wear the special goggles, their brains start receiving information that they don't know how to interpret. Over time their brains learn how to interpret that information. If they're like humans, the younger they are the quicker their brains would be able to adapt.