MQTT protocol works on pub-sub concept. Client can subscribe and publish topic to the Broker. Clients who are sensor kind of thing can publish respective topics (temperature, humidity, switch on/off request etc) to Broker. Broker will send the respective data to the subscribed clients (display device, any device which can on and off). Broker usually hosted on open internet.
I'm not getting the point how Broker will send the information to client device as the wifi connected device usually couldn't be reachable from open Internet as there will be no public IP assigned to such IoT devices. (I mean to say IP communication from Open internet to IoT device should not be successful)