To protect their identity when making phone calls using Twilio's API, developers can use proxy servers like Fiddler to mask their connection. By installing VS Code, .NET Core, and other tools, developers can create a C# application that makes phone calls with Twilio's API behind a proxy server. The application uses the Twilio NuGet package to authenticate and make calls, while the proxy server intercepts HTTP traffic sent to and from the machine it is running on, protecting the developer's identity. With the updated code, when the application runs, all web traffic, including calls to Twilio, flows through the proxy server, masking the developer's identity.