It's not 'Australian', just English.
Breaking it down:
'bill': A bill in a governmental context is a piece of proposed legislation. To become law (actual legislation), the elected members of a jurisdiction (paliament: government + (usually) opposition) will go through a process of reading the proposal, suggesting amendments, additions, deletions etc and sending it back to the legislation writers to make those amendments. This can happen a number of times.
'charter change': A charter is a constitution type document. Constitutions usually differ from normal legislation as they require a greater effort to alter. Changing the constitution of a jurisdiction is a significant and possibly controversial exercise and would be seen by most as relevant and newsworthy.
'third reading': As mentioned under 'bill', the parliament reads and amends the bill a number of times before making it law. The newspaper article is referring to the third time this bill went through the read/amendment process.
'sails through': 'Sails through' means 'easily successful. In context there were either no, or very few (and uncontroversial) amendments suggested during the reading process.
To understand as a whole:
"The government's proposal to change the constitution completed its third reading and amending process (recently). There was little to no opposition to the content of the (already twice-amended) proposed legislation during this reading."