It has no one particular meaning, but is generally used favorably; among other things, it has been used in reference to Australia's natural resources, weather, history, distance from problems elsewhere in the world, and other sorts of prosperity.
Australia is a lucky country, run by second-rate people who share its luck.
Horne's statement was actually made ironically, as an indictment of 1960s Australia. His intent was to comment that, while other industralised nations created wealth using "clever" means such as technology and other innovations, Australia did not. Rather, Australia's economic prosperity was largely derived from its rich natural resources. Horne observed that Australia "showed less enterprise than almost any other prosperous industrial society."
In the decades following his book's publication, Horne became critical of the "lucky country" phrase being used as a term of endearment for Australia. He commented, "I have had to sit through the most appalling rubbish as successive generations misapplied this phrase."