The ESP8266 sends SMS or MMS messages through AWS IoT, Lambda, and Twilio APIs by publishing messages to the 'twilio' MQTT topic, which triggers a Lambda function that extracts necessary fields and sends an SMS or MMS message. The ESP8266 connects to AWS IoT using MQTT over WebSockets, and uses a Twilio account for sending and receiving messages. To set up this application, users need to create a new Twilio account, purchase a SMS-enabled number, add the AWS IoT policy and IAM user, subscribe to the 'twilio' topic, and configure the Lambda function with Python code. The integration allows devices to send SMS or MMS messages using AWS services, enabling low-power IoT applications.