Ink is transparent; It lets through or absorbs light that hits it. Red ink lets though red light and absorbs the rest. Shine white light through it, and it will look red.
Shining white light on a black piece of paper with red ink on it will look black. What light is not absorbed by the ink is absorbed instead by the paper.
Of course red ink reflects light, as you say, but that is just because of reality. It is not essential to its function. If it were possible to make a 100% non-reflective ink, it would work perfectly well.
I hope this clears it up :)