Twilio, Bluemix, and AlchemyAPI are used to analyze the sentiment of text messages. By setting up a Twilio account and a Bluemix trial, developers can receive SMS messages and use the AlchemyAPI service to determine their sentiment. The application responds differently based on the sentiment, providing a positive or negative response. The project is built using Ruby and Sinatra, with the help of the Cloud Foundry CLI. Developers can push their local copy of the application to Bluemix and test it by sending messages to a Twilio number. The AlchemyAPI service provides a score and type for each sentiment analysis, which can be used to inform responses. By integrating these services, developers can build applications that understand and respond to user emotions in real-time.