Controlling Home Assistant on a Raspberry Pi with Twilio SMS involves sending an SMS to a specific number, which triggers a webhook that interacts with a web application hosted on the Raspberry Pi. The web application translates the command from the SMS content into a controlled object according to a protocol and calls the corresponding URL to the REST API of Home Assistant. With this system, users can control devices or objects in Home Assistant using SMS commands, such as turning on or off lights connected to GPIO pins. The process involves setting up Home Assistant on the Raspberry Pi, creating an entity for each device, and writing a Python script that interacts with the REST API of Home Assistant to turn the devices on or off based on the received SMS commands.