Twilio is ending its API explorer tool on December 15th, 2023, and will instead use Bruno, an open-source API client, as a replacement. Bruno allows developers to make requests to Twilio's REST API, explore responses, and copy code snippets for their applications without coding. To set up Bruno, developers need to import Twilio's OpenAPI Specification, configure the environment with Twilio credentials, and then use the API client to send an SMS without coding. The process involves setting up a Twilio account, phone number, and importing the OpenAPI Specifications into Bruno, configuring the environment, sending a test message, and exploring the API further.