I became aware of it as sixties counterculture word meaning "profound," in the sense of thought or philosophy. As Stuart Johnson notes, "deep" and "profound" are literally synonyms, and we speak of people being "deep in thought" when they are concentrating long and hard on something. To put it into a sixties context: "Marshall McLuhan says that the content of the new medium is always the old medium." "Really?" "Yeah, you can't make a distinction between 'the medium' and 'the message.' The medium _is_ the message." "Oh, wow. That's deep."

Besides the literal meaning of "deep" (for instance, I tell my friend "the middle of the Pacific ocean is 11 kilometers deep" and he replies "wow, that's deep") it figuratively means profound, or philosophically complex. Like if someone asks me what the meaning of life is, that's a deep question.

We usually says "that's deep" when somebody says or does something that we consider to be profound in some way.