Ok let me give this a try. The type of reaction this is called is endothermic. Its a big word simply to mean that it take energy/heat to make a something happen( a reaction).
For a solid to exist such as ice it has a very stable structure. The atoms of solids are very orderly and line up. To simplify imagine a cube with atoms in each corner. Now imagine stacking this cube with more cubes all around it. This is the fundamental state that gives solids is “solid feeling”.
Now if you add salt you are essentially trying to add another molecule into our ‘cube’ structure leading to less solid structure. It takes energy in the form of heat to make this happen. Heat comes from the surround environment ie air and water thus making it seem colder.