It is part of the constitution. What they're referring to is the events of 1975 when Gough Whitlam and the Labor party government failed to pass a budget by the hostile senate, leading to the dissolving of parliament by the Governor General.

Basically the Governor General is the Queen's official representative in Australia and officially our head of state. They have the power in specific circumstances to dissolve parliament and force an election. One of those circumstances is if the government fails to legislate a budget. It's complicated how the scenario occurs, but it happened, the powers of the governor general were exercised and a new election was held.

In this case the scenario is only possible because we recognise a power higher than our own parliament, that of the Queen. The USA doesn't have that but perhaps an alternative system could be worked out (if it's even desired, I think it's a useful thing to have).

[1]: http://en.wikipedia.org/wiki/1975_Australian_constitutional_...