Salt makes ice colder because of freezing point depression. Water freezes at a lower temperature because salt acts as an impurity.
For water, which is in liquid form to become ice or solid, energy needs to be removed from the water. Water molecules need to create bonds for becoming solid. The presence of salt or impurity makes it more difficult for bonds to form, resulting in a lower freezing point. Pure water freezes at 32 degrees Fahrenheit. The addition of salt makes water freeze at a lower temperature.
The science of salt making ice colder is applied in making ice cream. This science is also evidenced in sea water freezing, which is affected by salinity.