The fridge is connected to your water supply. It fills a mold/ice tray with water, the mold can come in many shapes and forms, it could look just like an ice tray. There is a sensor at the mold that detects when it's full and then closes the valve so the water doesn't overflow. That mold gets cooled down to freezing. Then move the ice from the mold to the storage bin. Once the storage bin is full it will stop the ice-making process. If the bin isn't full it will make more ice.
The cooling works the same way as your fridge/freezer. You have a liquid (refrigerant) that evaporates at a relatively low temperature at low pressure. Any evaporation process draws out heat. That's why sweating cools us down and why water feels cool even if it's the same temperature as air. Then you just build the system in a way that the evaporating gas is in a place where it will draw out the heat from the freezer. Then you send that now cold low-pressure gas through a compressor that turns it into a hot high-pressure gas. You slowly cool that gas down, once it gets cold enough it will turn liquid again. Then you send the liquid through an expansion valve to reduce the pressure and you are starting at the top again with a low-pressure liquid that evaporates and sucks out the heat from your freezer. Some fridge models will have a single cooling system for the fridge, freezer, and ice maker. While some other models have a separate cooling system for the freezer and fridge compartment. And you need a mix of high-pressure and low-pressure areas to ensure the evaporation process is always happening in the area where you want to cool down.
The sensors used in the ice maker are usually a mix of physical ones and temperature ones. The ice storage bin often just has a lever near the top that ice will push down once it's stacked high enough. So once you take out some ice, it will drop lower and the lever is released. But a thermometer near the top of the bin can also work. As the ice stacks higher the temperature near the top gets colder. And eventually, the thermometer will detect the temperature it's set to deactivate the ice-making process. The tray usually uses a temperature-based sensor to detect when the water is frozen.
Moving the ice from the tray/mold to the storage bin works in a lot of different ways. Often, the tray will briefly heat up so the ice isn't stuck to it, and then use gravity or some kind of motor to move the ice from the tray to the storage bin.