Straight from wikipedia:
The idea of cultural cringe was defined by Australian sociologists Brian Head and James Walter as the belief that one's own country occupies a "subordinate cultural place on the periphery", and that "intellectual standards are set and innovations occur elsewhere". As a consequence, a person who holds this belief is inclined to devalue their own country's cultural, academic and artistic life, and to venerate the "superior" culture of another country
So we dismiss our own culture and perceive other cultures to be more superior. In the case of language I'd say it refers to the Australian vernacular. So words that are 'typically' Australian. Such as drongo.
I personally cringe when I hear someone say g'day.
Then again I don't do English Language, so feel free to correct me.