The process occurring here is osmosis. When two solutions are separated by a semi-permeable membrane, water moves from a region of low solute concentration to high solute concentration.
The skin of raisins is the semi-permeable membrane. When the raisins are kept in water, the solute concentration is lower outside of the raisins and water moves in and the raisins swell.
When kept in salt solution, the solute concentration of the salt is almost equal on both sides and no swelling occurs.