An MQTT broker can be housed on your own LAN and often a RaspberryPi is used for that, or a public broker can be used.
Now obviously, just building one Node is not going to do much good. Who is it going to talk to? So obviously this would fit in a bigger systems with various nodes talking to eachother and the user having a frienly interface such as OpenHab. But in the google playstore various MQTTdashboards are available as app that allow for a simple interface. When developing MQTT software, it is always smart to install mqtt-spy. It is available for several platforms and what it basically does is to show MQTT messages going around on your LAN and to allow you to manually send MQTT messages