The Authy Phone Verification API is used to verify user phone numbers in Ruby on Rails applications. The process involves the user entering their phone number, country code, and verification method (SMS or voice call), which triggers an API request to initiate the verification. Once verified, the user is presented with a 4-digit code to enter, which is then sent back to the API for verification. If successful, the user's phone number is marked as verified, and they are redirected to a success page. The application uses the Authy JavaScript form helpers to augment the country code select field into a searchable list of countries.