This is actually a quotation from Dr. Timothy Leary that became a catchphrase and a slogan. Leary was a controversial figure, a clinical psychologist who advocated the use of psychedelic drugs. He taught classes at Harvard University, and he and Harvard were one of the centers from which psychedelic culture spread.
"Turn on" means something like "take psychedelic drugs, and become enlightened and energized."
"Tune in" means something like "resonate with the universe."
"Drop out" often means "leave school without graduating." In the context of the slogan, it means something like "disengage, separate yourself from conventional society, and become self-reliant."