The majority supported passing the bill in the House of Representatives (in parliamentary jargon, they supported reading the bill for a third time). It will now be sent to the Senate for them to consider.
The main idea of the bill is to recognise local governments in the Australian Constitution but it can't do this on its own. To amend the Constitution, there must be a successful referendum (as required by Section 128).
Unlike Labor Party members, Liberal members are allowed to rebel. However, it is increasingly uncommon.
What does recognising local governments mean?
Recognising local governments in the Constitution would mean that the federal government could directly fund local governments instead of having to fund them through state governments.
For an academic discussion of the effects of recognition, see Professor Anne Twomey's report (906 KB).
Background to the bill
See the bills digest (775 KB) for more background information.