The reason is due to osmosis, or the movement of water through a semi-permeable membrane. In this case, the outer skin of the grape is the membrane, thus separating the inner pulp of the grape from the surrounding solution. During osmosis, water tends to naturally traverse from an area of high concentration to low concentration. Since the hypertonic salt solution that the grape is sitting in has lots of dissolved solute, this means that the water concentration is higher inside the grape than outside. So the water leaves the grape through the skin into the hypertonic solution to try to balance out the concentrations. As a result of this loss of water, the grape will lose mass and shrivel up. This type of fruit drying is how grapes are turned into raisins.
A grape will shrivel up in a hypertonic solution of salt due to osmosis, the diffusion of water. Osmosis is a process where water goes from "high to low." This means, in order to achieve equilibrium, water flows from a high concentration to a lower concentration. Because there is a lower concentration of water (and a higher concentration of solute/salt) in the hypertonic solution, the water in the grape will flow outwards.