The Nexmo Verify API allows developers to validate users' phone numbers, ensuring that they are indeed reachable on the provided number. To set up this process in Node-RED, developers need a Vonage API account and credentials, which can be found on the dashboard. The `sendverify` node is used to start the verification process by sending an SMS message with a PIN code to the user's phone number. After submitting the received PIN code, the `checkverify` node compares it to the one generated by the Verify API, and if successful, outputs a response indicating that the verification has been completed. The status of the verification can be used to customize notifications to the user, such as displaying a success message or an error message if the verification fails. Additionally, developers can implement buttons to escalate the verification process to a TTS call instantly or cancel the verification process altogether. The Nexmo Verify API Reference provides more information on the available fields and their meanings in the response objects.