Salt wants to mix with water to form a solution, but in order to mix, the ice needs to melt. In order to melt, the ice needs to take heat energy from the environment around it. Salt lowers the freezing temperature of ice, meaning it can get colder than 32F (0C), and still stay liquid. It also means it will melt faster in temperatures above freezing.

As more ice melts, it takes more energy from the environment, which lowers the temperature inside the can to below freezing. This happens inside of the ice + water can too, but without the salt, it can only lower the temperature to just above freezing.

Meanwhile, something else is happening on the outside of the cans. 

The air around us is made of many different gases. One of those gases is water vapor. Do you know what happens when water vapor (H2O in gas form) cools? It turns into a liquid, and then a solid.

The air inside of the can is colder than the air outside of the can, so water vapor gathers and forms dew on the outside.

Since the salt + ice can is below freezing, that dew freezes and turns to frost!