I don't know about this app specifically, but I can answer based on the usual behaviour of music players. If music is playing when you leave the app the usual way, the app starts a foreground service to keep playing the music. This is a special kind of service: it shows an "Ongoing" notification and Android will not kill it to free memory, unless it's really desperate.
Sometimes, this is exactly what the user wants: you start playing music, switch to another app, and the music keeps playing. But sometimes, it's not clear how to make it stop, and avoid the foreground service taking up memory. The "Close" option gives you an obvious way to do this.
In most cases, app authors shouldn't add a "Close" option, because it confuses users (just as you've been confused here), and because the Android app lifecycle doesn't work that way. In this case, the potential for confusion caused by the foreground service is greater, so it's a sensible trade-off.