A phonetic spelling of Australia as pronounced in a broad Australian accent. The word Straya is pronounced something like "stray-yah".
Depending on the context, the word might imply an affection for something seen as peculiarly Australian in nature; or contempt for something seen as peculiarly Australian in nature. It often serves as shorthand for the phrase "only in Australia". According to the rules of Australian language and culture, you can distinguish as follows:
Affection (with a fair amount of embarrassed pride) (phrased as contempt, because that's how we do it):
"Bloody emus, yeah, we had a war against 'em once. We lost. Straya!"
Contempt (disguised as politeness, because that's how we do it):
"Another PM rolled. Better check the batteries on your smoke alarm. Straya!"
Although the word has been in use for as long as there has been an Australian accent, the development of a specific spelling and usage is more recent - perhaps within the past twenty years.
See also bogan, Strine, Murica, Merkin.