The Telesign SMS Verify API is a service that allows developers to send verification codes via text message, which can then be used to verify user identities. The process involves five steps: getting an API key, registering for the Telesign SMS API, creating a registration form, integrating with the API, and putting it all together. The API provides three input parameters, including the recipient's phone number, verification code, and app name. Once the registration form is submitted, the API sends a verification code to the user's phone via text message. The user must enter the verification code within 15 minutes or they will have to try again with a new code. The process also includes storing the user's information in a database and tracking attempts to prevent brute-force attacks.