how exactly does one save money by using a smart thermostat?
Short answer: By knowing when nobody is home it can save energy by not heating/cooling at those times.
Basically, there are 3 classes of thermostats:
Basic - Set the temperature whenever. It wastes energy whenever nobody is home, when they forget to set it higher/lower as needed before they leave.
Programmable - Program in a schedule. It wasted energy whenever people go off-schedule. For example, if programmed for the AC to be at a nice and economical 85 during the day and then down to 72 at 6pm when people get home from work, it will still kick on to 72 even if the people are away on vacation or decided to go out for drinks with friends after work.
Smart - Programmable, but also senses occupancy. So it knows that if nobody is home to just use the economical settings regardless of what the schedule is.
How can set a particular room to be a certain temperature unless each room has its own AC/Heating unit.
The sensors aren't for setting particular room temperatures. If you set your thermostat to 72, that doesn't mean all the rooms are at 72. Some rooms have more or less windows, some rooms have better or worse insulation, some rooms are above the garage or below the attic. In other words, when the thermostat is set to 72, the area directly next your thermostat will be 72 but the other rooms can be anywhere between 68 and 74 or so. Some people will find 68 too chilly. Some people will find 74 to be too sweaty. Your thermostat alone cannot know what the temperature is in the other rooms. That's where the sensors come in. They know the temperature of the other rooms
and whether there are people in that room. So if the thermostat is set to 72, it will work to keep the rooms where people actually are at 72. This might mean the room where the thermostat actually is actually at 68 or at 74, but it doesn't matter. What matters is the room where people are that moment.