The process is fairly simple. The system uses a small computer (the ESP8266) to make a google API call every 10 seconds requesting the amount of subscribers for a given channel. (You can specify what channel in the code). YouTube responds with the count and then the LED display is updated with this count! Simple right!? If internet connection is lost or fails, it just keeps trying every 10 seconds.
It then updates the display and changes the colour based on growth, maintaning, or reduction. You can find more details about the Live Sub Count for my YouTube Play Button here.