The Vonage Voice API allows for easy integration with ASP.NET applications, enabling features such as making and receiving phone calls. To set up this functionality, developers must create a Vonage application, generate public and private keys, and configure their ASP.NET project in Visual Studio. The `appsettings.json` file is used to store sensitive information like the Application ID and Private Key path. A new controller called VoiceController is created to handle business logic for making a Text-to-Speech phone call, which uses the VonageClient to create a call with specified endpoints and credentials. Once set up, developers can run the app and make a successful Text-to-Speech phone call.