The Vonage Voice API allows developers to build high-quality voice applications in the Cloud, enabling them to create interactive voice response menus like this tutorial demonstrates. The process involves creating a Vonage Application and virtual phone number, using ngrok to expose webhook endpoints on a local machine as a public URL, setting up a Node.js project with Express and body-parser to handle incoming requests, defining endpoints for Answer and Event URLs, handling user input via DTMF, and running the application by executing the `node index.js` command.