With the help of the Vonage SMS API, developers can send SMS messages anywhere in the world using their virtual phone number, Ruby SDK, and a few lines of code. To integrate Nexmo into a Rails application, one needs to set up the basics, including obtaining an API account, installing the Nexmo Ruby Gem, and initializing it with credentials. The integration allows for sending SMS messages from within the application, storing the data in a model, and then sending the message using the Nexmo API. The process involves creating a form, storing the SMS record, and then sending the message using a method that updates the message ID on the SMS record if the status equals 0.