The main reason for this instruction is to speed up the rate of diffusion. Neutralization reactions are determined by the end point. In theory, an endpoint can be determined from a slope in a graph. However, the slope is very steep, such that a slight change in addition of titrant, it would exceed the endpoint. Therefore, you have to stir the solution so that you can see the immediate reaction with every tiny drop that is added. There might be cases where the color is not significant because it still hasn't diffused in the solution very well.