The first order of business will be to flash the provided Arduino sketch to the ESP8266 with the Arduino IDE. Download the IDE for your platform and follow these instructions to add support for the ESP8266 platform.
Next you will need to add the PubSubClient library for MQTT. Follow this guide for the general process and add the 'PubSubClient' from the Library Manager. Once that is installed you will need to edit the PubSubClient.h file and change the line #define MQTT_MAX_PACKET_SIZE 128 to #define MQTT_MAX_PACKET_SIZE 512. You can find the installed library under the path shown in File > Preferences > Sketchbook location.
At the top of the Arduino sketch are several fields you must modify to fit your environment (WiFi details, MQTT broker IP, node name, etc). Once those fields have been set you can upload to your microcontroller and start sending commands to the display.
If you will be deploying more than one of these devices you must change the node names to be unique. Failure to do so will result in a cascading series of MQTT connections/disconnections as your devices compete for access to your broker.
After the initial Arduino deployment via USB you should be able to apply future updates to the ESP8266 using OTA updates, allowing you to update code while the device is installed.