It looks white because the light bouncing off the 3 patches is being added together when you spin it. Your eye is seeing a flash of red, then green, then blue, then red and so on as it spins. In other words, the red, green and blue light is summed by your eye over a very short time as the paper spins quickly.
This is not same as mixing red, green and blue paint pigments together. In that case, the red paint absorbs blue and green colours, the green absorbs red and blue colours and the blue absorbs red and green colours. Stirring the paints all together winds up absorbing all the colours, so you get a very dark, almost black, mixture.
If you had 3 pieces of red, green and blue transparent plastic, then you could try shining white light through them in two ways. First, with 3 flashlights and the 3 coloured plastics you could shine red, green and blue light onto a single spot and get white light. This would be like the spinning paper case. Second, you could shine a single flashlight through the 3 plastics stacked on top of one another. You'd see that very little light makes it all the way through all 3 plastics. This is similar to the paint mixture.