The Nexmo SMS API uses HTTPS protocol to send messages, which involves a TCP handshake, TLS encryption, and a POST request to the server. The cURL command-line tool is used to send the request, which includes parameters such as API key, sender ID, recipient number, and message content. The output of the request shows the DNS lookup, TCP connection establishment, TLS handshake, and server response headers and body. The response body contains information about the sent message, including the message count, status, balance, cost, and recipient network ID.