Twilio requires an integration between the SmartThings Cloud and the Twilio cloud. The resulting device in SmartThings would be one that supports the “Notification” capability. SmartApps would then send text to this ST device, which would result in the Twilio cloud being notified, and ultimately Twilio would send you a SMS text messsage to your phone.
At least, that is my understanding. IIRC, the problem with this design is that SmartThings doesn’t really seem to support using Notification devices in their Official SmartApps, instead relying on Push Notifications to the SmartThings App on your phone (or SMS…) If you only send messages from webCoRE, then it should be simple to use a service like Twilio or even Pushover.