The Vonage Number Insight API provides valuable insights about phone numbers, including validity checks and information about the network provider, country, and international dialing options. To use the API, developers need to install the Go SDK and create an account with a Vonage API key and secret. The API offers three levels of insight: basic, standard, and advanced, each providing more detailed information about the phone number. Basic insights provide general information, while standard and advanced insights offer more detailed data on network providers and international dialing options. Advanced insights also include additional information such as the name of the person associated with the phone number, but require an asynchronous request and a webserver to receive the response.