The act made Australian law independent of the British legislature and judiciary. It was in fact the final one of seven acts created by the seven Australian parliaments required for such a constitutional change to the whole Federation. The parliament of the United Kingdom also passed an act to complete the process.
One of the most significant powers of the act was the termination of appeals from Australian courts to the Judicial Committee of the Privy Council in London. The act removed the Privy Council from the remaining "direct lines of appeal" in the Australian legal system (particularly in state legal systems) placing greater emphasis on the High Court of Australia as the final court of appeal in Australian law.
Prior to this act, the "Privy Council (Limitation of Appeals) Act of 1968" made the High Court the final court of appeal on both Constitutional and Federal Matters. The exception to this was inter-se constitutional matters, which are still permitted to appeal to the Privy Council if the court issues a certificate under section 47 of the Australian Constitution.
The "Privy Council (Appeals from the High Court) Act 1975" removed appeals to from the High Court to the Privy Council in both state and common law matters.
So with these two acts in place, nearly all appeals from the High Court to the Privy Council were removed. However, direct appeals from State Supreme Courts remained possible until the passing of the Australia Act, and it's associated State Legislation.
A final remaining appeal to the Privy Council can be achieved by a certificate issued by the High Court. This is because the Australia Act, although pivotal in furthering the sovereignty of Australia, could not alter the Australian Constitution (a national referendum is required to do so.) However, the High Court has made it quite clear on numerous occasions that it will never grant such a certificate.
Another function of the Australia Act was to establish some form of independence to the States of Australia. Legislation such as the "Statute of Westminster 1931 Act" granted the Commonwealth of Australia various degrees of nationhood, but failed to apply to the State Governments. Because of this, prior to the passage of the Australia Act, these states essentially remained colonies under the British Crown.
It's because of this emphasis on the states of Australia, that each state parliament was required to pass it's own legislation. The long title of these "state acts" were "An Act to bring constitutional arrangements affecting the Commonwealth and the States into conformity with the status of the Commonwealth of Australia as a sovereign, independent and federal nation." Collectively, the federal, state and British acts were known as the "Australia Acts."