Using phone app to control remote devices

I have been hungry to have a layman's understanding of how a smartphone app can, while connected to the internet, remotely connect with, control and/or change the program on a wifi connected device, i.e. lights, thermostat, alarm system. I think I understand how this is possible if the phone and appliance are using the same wireless network within a home, but how is it possible if the phone is on the other side of the country? Is the phone really talking directly with the appliance, and how? And what enables the app to connect with a specific device? Broad questions, I know. But I'd sincerely appreciate any insight. Thank you.