Stripe SMS notifications are being sent through Twilio and Python, using Flask as the microframework for handling incoming POST requests. The application is deployed to Heroku and configured with a webhook to receive Stripe payments successfully, sending an SMS notification to the store owner's phone number. The process involves setting up a new Git repository, installing dependencies, creating a Procfile and requirements.txt file, initializing and committing all files, deploying the app to Heroku, configuring config vars for the app, and testing the finished app with Stripe.