AWS Lambda is a serverless service that can run Python 3.6 code in response to developer-defined events, such as uploading a new JPEG file to AWS S3. To send SMS text messages via Twilio, developers can create an AWS Lambda function with a manual test event trigger and use the Twilio SMS REST API. The process involves creating a Lambda function, writing Python code to handle the SMS request, setting environment variables for Twilio credentials, and configuring the Lambda function handler and role. Once configured, the Lambda function can be tested by simulating an event, which will send an SMS message to a specified recipient phone number.