I know that when you put salt on ice it may turn to a slush and it does feel cold. What is the reason for this?
State of matter changes from solid to liquid and from liquid to gas are endothermic reactions. When you sweat, water evaporates and that cools your body. You don't have to "put the energy into it" (intuitively as in heating up your skin over 100 degrees Celsius) for the state change to occur if it occurs spontaneously and the molecules in changed state are captured in that state, in the case of sweating by leaving the proximity of the liquid droplets on your skin (which is also why fans work to cool you even if the ambient temperature is higher than your body temperature).
Now back to your case of adding salt to slush it's the same thing: you have the salt holding the spontaneously molten ice in liquid state, and taking up heat in the process.
Conversely, when you have a cold beer can in a humid weather, the condensation drops forming on the can will heat your beer can quicker than if you were in super dry weather in the desert.
Oh, there is also solution enthalpy, which in case of NaCl is negative, i.e., dissolving NaOH is also endothermic. If you threw in NaOH instead, your slush might actually warm up.