Workflow

First understand the principle, then test further with MQTTx, and finally implement it on the ESP8266. Following this order means that if you hit a problem, you can track it down right away.

Check Whether the Firmware Works

AT+MQTTUSERCFG=0,1,"ESP8266Client","myname","password",0,0,""
If it doesn't throw an error, it works.
If it errors out, that means the firmware doesn't support MQTT — please flash the following firmware:
All the rest of this series is explained based on that firmware. If you really don't want to use the firmware below, please don't watch this video.