When performing chromatography, it is necessary to find a solvent that will dissolve the pigment in question. Water can dissolve polar solvents, but it is very poor at dissolving polar solvents. In contrast, a fully nonpolar solvent such as mineral oil would have the opposite effect.
Also, the solvent must be able to travel up the paper to allow separation of pigments. The further up the paper a solvent can travel, the greater amount of resolution can be achieved. Thus, polar solvents, especially water, travel more slowly and allow less resolution between pigments.
Acetone provides a great middle ground for this process because it is amphipathic. An amphipathic substance has both a polar end and a nonpolar end. Looking at the structure of acetone, you can see that it has a significant partial negative charge on the oxygen atom, a slight positive charge on the carbonyl carbon, and it has two nonpolar alpha-carbons. It is, therefore, polar but less polar than water, and it has the capability to dissolve nonpolar substances, as well. Its slight polarity allows it to dissolve polar substances, and the fact that it is less polar than water allows greater resolution between pigments on paper.
These reasons allow acetone to be a great solvent for pigment chromatography. Other solvents that are good are small alcohols for the same reason!