The distribution coefficient is a measure of how well a compound dissolves in one solvent relative to another. In this case, the distribution coefficient of caffeine in chloroform and water is 10, which means that caffeine is 10 times more soluble in chloroform than it is in water.
If the goal is to extract 90% of the caffeine present in one extraction, this means that 90% of the caffeine must be dissolved in the chloroform, while only 10% remains in the water. Since the distribution coefficient is 10, this means that the relative volume of chloroform-water must be 10 times larger than the relative volume of water.
In other words, if the relative volume of water is Vo/Vaq, the relative volume of chloroform-water must be 10 * (Vo/Vaq) in order to extract 90% of the caffeine. Therefore, the relative volume of chloroform-water that must be used is 10 * (Vo/Vaq) = 90% of the caffeine.