Embedding Elastic App Search into Twilio Flex allows agents to efficiently find information, reducing their time spent searching for data across various platforms. To achieve this, developers create a Flex plugin using Search-UI, a React library by Elastic, and deploy it on Twilio hosted Flex. The solution involves setting up an App Search cluster using Docker and Docker Compose, creating an engine to host documents, and obtaining credentials to access the search interface. With the plugin deployed, agents can quickly find relevant information, improving their productivity and customer experience.