A drop of ink is spotted just above the bottom edge of a paper strip, which is then stood in a shallow pool of solvent. The solvent soaks up the paper by capillary action, its front climbing roughly with the square root of time (Lucas-Washburn behaviour).
Each pigment in the ink competes between clinging to the paper fibers (stationary phase) and dissolving into the moving solvent (mobile phase). A pigment that dissolves easily rides high with the front; one that clings to the paper barely moves. That balance is fixed for a given pigment/solvent pair as the retardation factor Rf = (distance travelled by the spot) / (distance travelled by the solvent front), always measured from the original spot.
Raising solvent polarity favours pigments that are themselves more polar, pushing their Rf up and pulling non-polar pigments' Rf down — which is exactly how a chemist tunes a solvent to resolve a stubborn mixture into clean, separated bands.