Our sky looks blue because the molecules in the air scatter the white light coming from the sun, and the blue light is scattered more than red light.
But in space, there is no air, so none of the light coming from the stars gets scattered. We see bright points of light of the stars. but everything else inbetween is empty – nothing to scatter the light, so it looks black.
Space is only really black to our incredibly crap light detection systems (our eyes).
Like how the Hubble deep field shows that even the darkest patch of sky is actually full of light. In fact, the entire sky is brimming with light, it just happens to have been redshifted (changed in wavelength) so far that the light is now microwaves that we can’t see with our puny eyes. That’s the cosmic microwave background – if we could see microwaves, everywhere we looked we would be able to see an image of the Big Bang!
Unfortunately our eyes, and most cameras, can only see things if enough photons come from them to set off the little detector. That may only be one every second (I don’t know the actual number) but if something is fainter than that, we won’t be able to see it.
So, we can see stars close by, because they put out enough photons to set off our detectors. We can see things close by that reflect photons because, again, there are enough to set off the detectors.
If we looked at the darkest patch of sky, there’s probably something like a million stars, right there. But they are so far away, their light so spread out (light spreads out as it travels) that there aren’t enough photons hitting out eye.
The only way to see the real colour of space would be to have an infinitely long exposure, pointed in exactly the same place for ages. If you could even manage to do that, the motion of the Earth, the motion of the Sun through the galaxy and eventually the motion of the galaxy from the universe would spread out the light from the stars like an afterimage.
If, and that’s a big if, if we could get rid of all that, Space /wouldn’t/ be black. It would be an amazing coloured masterpiece, the greatest of its kind.