When you ask why, you want to know about causality. If I ask "why does the cold pack show a decrease in temperature" and the answer is "because the reaction is endothermic", this might be considered a tautology. After all, endothermic means that energy is needed, and this energy can come from the surrounding, lowering the temperature.
Why does ice water get colder when salt is added?
As the OP states, this lowers the freezing point of the liquid. The system is no longer at equilibrium, and some ice will melt in an endothermic process. As a consequence, the temperature drops and the salt water gets diluted. The melting process stops when salt concentration and temperature are matched again, i.e. the freezing point of the liquid is equal to the temperature of the system.
It is well known that when you add salt to ice, the ice not only melts but will actually get colder.
The melting process is at the interface of liquid and solid, so both the solution and the ice will get colder.
From chemistry books, I've learned that salt will lower the freezing point of water. But I’m a little confused as to why it results in a drop in temperature instead of just ending up with water at 0 °C.
So the question is given that some ice melts, why does the temperature drop. Saying that ice melting is an endothermic process maybe does not fully answer the question (explain the causality).
What is occurring when salt melts the ice to make the temperature lower?
In terms of kinetics, the salt does not melt the ice. Instead, it lowers the rate of water freezing. The net effect is that ice melts. At the molecular level, according to https://www.nyu.edu/pages/mathmol/textbook/info_water.html, "In liquid water each molecule is hydrogen bonded to approximately 3.4 other water molecules. In ice each each molecule is hydrogen bonded to 4 other molecules." So upon melting, water loses about half a hydrogen bond. Also, the remaining hydrogen bonds might have less ideal distances and angles. So that's what makes the process endothermic. The NaCl has little role in the energetics, as any other solute has pretty much the same effect (colligative property).