"How does the Commonwealth Constitution create a third level of Government?"

Great question! Technically, it doesn't.

Local councils are not mentioned in the Australian Constitution but each state has a Local Government Act that provides the rules for the creation and operation of councils.  While these Acts vary across Australia, in general they cover how councils are elected and their power to make and enforce local laws, known as by-laws. A by-law could be considered a form of delegated legislation because the state government gives, or delegates, to councils the authority to make laws on specific matters.  As councils derive their powers from state parliaments, council by-laws may be overruled by state laws. Australian territories have arrangements covering local government responsibilities; however these arrangements are unique and not defined by the Constitution.

When an aspect of our system of government is not mentioned in the Australian Constitution, it does not necessarily make it unconstitutional.  For example, the Prime Minister is not mentioned in the Constitution but the position is an accepted part of our federal government structure.

In fact, the Constitution does not cover all aspects of the governance of Australia. While the constitution is central to the way government works, customs and convention also play an important role.  

diagram to help explain How does the Commonwealth Constitution create a third level of Government?