In the "How to Get CPU Information in Linux" YouTube video, the speaker shows how to access CPU details using the proc directory, a virtual file system that stores system information without occupying hard drive space. By typing "ls -l" in the proc directory, one can view various read-only files, including "cpuinfo." The speaker then uses the "cat" command followed by "cpuinfo" to display comprehensive information about the CPU, such as its manufacturer (Intel), generation (12th), model (i5-1240p), number of cores (two), and CPU frequency and cache size.