Humans have three cones of color light, red, green and blue. If the region of the green cone is hit by his photons we will see green light. Now we don't really receive green light but our eyes and brains are 'making' it to look green. But if all the different cones are hit by their photons we will see white. Now the question is why is that white? Why isn't our brain just keeping these different colors, is this a kind of black out of our brain or an overwhelming of input making us blind etc?
Our brain doesn't really use cones to detect green light, and detect red light, and detect blue light. Each cone can be excited by many wavelengths; they're just most excited by green, red and blue wavelengths respectively. The brain then uses the different information it receives from all three cones to infer the wavelengths of the light it receives. It doesn't look at the absolute excitation of a cone so much as the relative amounts of excitation different cones get.
So for example, if the brain sees the green cones are very excited and neither the red or blue cones are, the light is green. If the green cones and red cones are equally excited but the blue ones aren't, then the wavelength may correspond to yellow (a wavelength that excites both red and green cones in the same middling way, and blue cones not much at all), or there could be a combination of green and red wavelengths; and in fact both are seen as the same thing by the brain. That's why colors can combine to make other colors.
This is OK insofar as the point isn't to calculate the exact wavelength of incoming light, but to tell the difference between certain wavelength compositions because that provides more information about the world.
So the reason we see white when hit with all wavelengths is the same reason we see any color when hit with any combination of wavelengths.
As for why "color" feels like a single unified perception and not a combination of three numbers, that's a general aspect of perception in general. We don't sense the raw data our brain gathers; we sense how the world is (as guessed at by the brain from the raw data it gathered).
As such, the colors we see aren't even strictly determined by the wavelengths of photons hitting the cones; because colors are a proxy for what the things we see are like, the brain will not only look at the light it receives, but also use its own assumptions about the world (is this thing in shadow, in bright light, mottled light, colored light, near, far...) to guess at what the color "really" is given the wavelengths it has received. This gives us a sense that color is inherent to objects and doesn't depend on lighting conditions, which is generally accurate (an apple's redness is a feature of the chemicals at the surface of its skin, and those don't change with the lighting conditions).
That's where illusions like that of the blue-black or white-gold dress come from: sometimes you get ambiguous stimuli, where the wavelengths hitting the eye could correspond to a black and blue dress in bright light, or they could correspond to a white and gold dress in shadow, and in that case the brain picks one, and you see the dress as the color your brain decided it must be. Except because the stimuli were ambiguous, different brains (or the same brain at different times) will make different guesses, resulting in very different perceptions. But the guessing is actually what happens all the time, it's just that usually the stimuli aren't as ambiguous so all brains agree on their guess, and the guess does correspond to reality, and so the illusion that we perceive the world directly with no filter remains unbroken.