To answer OP's original question, the action is simply mixing. What happens after that depends of the components in the system. In the discussion so far, two threads of thought got mixed up, so let me comment from the perspective of a Physical Chemist.
Start with what happens in ALL cases where a salt is mixed with ice. The core concepts here are dynamic equilibrium and a property called "Fugacity". With just pure ice present, there is a dynamic equilibrium between the solid and liquid phases of water. At all times there is a tiny film of liquid water on the surface of the solid phase. Simultaneously, some of that liquid will crystallize into solid phase, and some solid ice will melt. The phase change from liquid to solid releases enthalpy in the form of heat, and that same enthalpy is consumed by a bit of ice that melts to make liquid water. This happens all the time at any temperature. There always will be some liquid phase water present, although it might be a very small quantity at very cold temperatures!
Now, add some solid salt crystals - and from a Chemist's perspective, there are hundreds of compounds that are salts, and most of them are soluble in water. Some of the salt dissolves in the thin water film it contacts, producing a solution of salt molecules and ions in the water. Since there is very little liquid water in the vicinity the solution rapidly comes to saturation - that is, the maximum concentration of that salt that can be made in water at that temperature. Now, that creates a solution of higher Fugacity (think a little like energy state) than a dilute solution, and certainly higher than pure water that contains no solute molecules. To reduce the Fugacity (energy state) we need to remove the salt, OR provide more liquid water. Well, the salt is going nowhere. Equally importantly, the solution now has a new Freezing Point that is LOWER than that of pure water. So the dynamic equilibrium between solid and liquid phases of water has been disturbed. If any solid ice is to "steal" enthalpy from the liquid surrounding it to melt, the resulting liquid solution that gave up the energy gets colder, but not cold enough to change to a solid. So, pure water is added to the solution by melting some of the ice, BUT that water stays in the liquid phase and no liquid water crystallizes into ice. The result is a solution with more water and hence slightly less concentrated in salt content. Hence it can dissolve more of the solid salt crystals still present, keeping the solution saturated. This process of irreversibly converting ice solid to liquid water so that the salt solution can be diluted into a lower-Fugacity state will continue until all the solid salt crystals are dissolved. And remember that all that melting of ice is sucking energy out of the liquid solution, making it colder. But then it STILL continues, because adding more liquid water to the solution still reduces the solution's salt concentration, reducing its Fugacity. Hypothetically this could continue almost forever (as long as we don't run out of ice), except for an important limit. As the solution becomes more dilute, its Freezing Point will rise. Eventually we will reach a new dynamic equilibrium in which the temperature of the solution does match its Freezing point. From then on, any further melting of ice to liquid will successfully cool the liquid enough to cause it to crystallize just enough to match the amount of ice that melted. A new dynamic equilibrium has been established.
This process is at the heart of how Calcium Chloride can melt ice at temperatures below water's freezing point. That particular salt is really effective for two reasons. It happens that its impact on the freezing point is particularly strong, so the resulting solution (if there's lots of Calcium Chloride present) won't freeze until it gets REALLY cold. Plus, it happens that the process of dissolving Calcium chloride in water is an EXOTHERMIC reaction - that is, it is a type of reaction (involving the formation of associated complexes of ions with water molecules) that RELEASES heat - in fact, quite a bit. So in the initial times when Calcium Chloride is sprinkled on ice, the dissolution process releases heat into the system that accelerates the melting of solid ice to liquid water and gives the whole "disturbing the dynamic equilibrium" process an extra boost start.
Now, the salt and ice for chilling ice cream makers is a little different. In fact, calcium chloride is not such a good choice for this job because of its high heat of solution. Plain old table salt, sodium chloride, was used first and is still a good choice. The resulting solution won't ever get to the really cold melting point that Calcium Chloride can produce, but that temperature is not needed to freeze the milk in the ice cream maker. BUT the whole process of causing ice to melt and make a colder solution of salt in water still is working. To maximize the effect, you must insulate the whole machine so that no outside heat can enter and all of the heat required to melt the ice comes from cooling the solution. PLUS, you put in LOTS of salt so the solution never actually escapes the saturation stage because there is always more undissolved solid salt present to drive the process.