Nexmo's Number Insight API provides real-time intelligence about phone numbers, with three levels of information available: Basic, Standard, and Advanced. The API is used to determine the country where a number is registered, its local and international representation, and other details such as line type and Mobile Country Code. The Advanced API also provides additional data on the validity and reachability of the number, as well as whether it is likely to be valid or ported. To use the API, developers need to sign up for a Vonage account and install the Nexmo REST API client for Node.js, then send an input phone number to the server to retrieve the insight data. The process involves setting up a Koa.js project, installing dependencies such as `koa-router` and `koa-bodyparser`, and creating routes to handle incoming POST requests.