The application verifies phone numbers in Symfony 4 using Twilio's Authy API. It captures users' phone numbers during registration and sends a verification token via SMS. The user must enter the code to verify their number, and if correct, it saves the details into the database after ensuring that they are not false or fraudulent registrations. The application also includes features such as login, logout, and secure areas for authenticated users. It uses Symfony's security component to handle authentication and authorization.