In the "How to Work with TAR files in Linux" YouTube video, the speaker shows how to create and extract tar archives using Linux commands. The process starts with the tar command syntax, including examples of creating an archive and extracting all files or listing all files. The speaker then creates a "demo" directory, copies two files into it, and includes these files in the tar archive using the "tar c zF" command. The speaker utilizes the "tar -tf" command to view the archive's contents and extracts the files to a specific directory using the "tar -xvf" command with the "C" option to specify the chain directory.