The Oxford English Dictionary cites uses of the meta- prefix as "beyond, about" (such as meta-economics and meta-philosophy) going back to 1917. However, these formations are directly parallel to the original "metaphysics" and "metaphysical", that is, as a prefix to general nouns (fields of study) or adjectives. Going by the OED citations, it began to be used with specific nouns in connection with mathematical logic sometime before 1929. (In 1920 David Hilbert proposed a research project in what was called "metamathematics.")
A notable early citation is Quine's 1937 use of the word "metatheorem", where meta- clearly has the modern meaning of "an X about X". (Note that earlier uses of "meta-economics" and even "metaphysics" do not have this doubled conceptual structure – they are about or beyond X but they do not themselves constitute an X).
Douglas Hofstadter, in his 1979 book Gödel, Escher, Bach (and in the sequel, Metamagical Themas), popularized this meaning of the term. The book, which deals extensively with self-reference and strange loops, and touches on Quine and his work, was influential in many computer-related subcultures and is probably largely responsible for the popularity of the prefix, for its use as a solo term, and for the many recent coinages which use it. Hofstadter uses meta as a stand-alone word, both as an adjective and as a directional preposition ("going meta", a term he coins for the old rhetorical trick of taking a debate or analysis to another level of abstraction, as when somebody says "This debate isn't going anywhere"). This book is also probably responsible for the direct association of "meta" with strange loops, as opposed to just abstraction. The sentence "This sentence contains thirty-six letters," and the sentence it is embedded in, are examples of "metasentences" that reference themselves in this way.