The crystals are soluble in water, so that near the crystal the water will have a higher concentration of the solute. Because the concentration is higher in the region, the solute ions will randomly move (diffuse) to regions that have lower concentrations - thereby spreading the solutes. This process will continue until the solubility limit is reached or the solid phase is no longer present and the concentration is constant throughout the liquid phase.
Equilibrium is reached faster if the temperature is hotter because the diffusion rate is faster as the molecules move around at a greater average rate.