This term is believed by American's to be a common phrase in Australia. This was originally due to an Australian tourism advertisement starring Paul Hogan that was aired in America from 1984 - 1990.
Barbie is a slang term in Australia for barbeque, alternatively known around the world as a barbecue or BBQ.
Although shrimp are known as prawns in Australia, the wording of the advertisement was designed to minimise confusion in American audiences.
In the 2005 American film "Dukes Of Hazzard", Bo Duke spouts "chuck another shrimp on the barbie" (a common misquote of slip another shrimp on the barbie) in a poor Australian accent when introduced to an Australian university student in Atlanta.