Solubility Vs Temperature
Solubility of any salt is a function of both Pressure and Temperature. Generally it is a weak function of Pressure while it is a strong function of temperature. I think by exothermic salt you mean salts having a exothermic nature of dissolution.
Supersaturation of a salt solution in any solvent is a relative term. It is brought upon by dissolving more salt than the saturation solubility requires by means of heating. Heating increases the temperature and it in turn increases the saturation solubility.
This can be viewed in preparing sugar syrup. Sugar is dissolved in little water and heated till all the sugar crystals dissolve and further concentrated i.e. water is evapourized to the required consistency.
Exothermic salts by dissolution increase the temperature of the solution and thus increasing the saturation concentration of the solute. Further heating will generally will increase saturation concentration and the solution can take more salt to reach saturation. That is why the exothermic salts can be used to prepare supersaturated solutions.