You have to understand that it happened in the context of Australia's history of Protestant-vs-Catholic religious sectarianism, which it inherited it from the UK and Ireland, and although by the 1960s and 1970s that was already fading (albeit still not yet completely dead), many alive then could remember earlier decades when it was much stronger. You also have to understand that, prior to World War II, the vast majority of Australia's Catholics were Irish, and so the religious tensions between the Protestant majority and the significant Catholic minority (>20%) often had overlaid on them ethnic/national tensions related to Ireland's fight for independence from the UK.

In the 19th century, Australia's colonial governments (which later became its state governments) provided public funding of private religious schools, both for the Protestant majority and the Catholic minority. In the second half of the 19th century, there was a big push for secular government education. Although that in itself might be seen as a religiously neutral idea, in practice it was something most Protestants were happy to support, the Catholic Church opposed. The result was that Australia ended up with a government school system which while officially religiously neutral, was de facto perceived by many as Protestant; most Catholics continued to send their children to Catholic schools, albeit now without any government support.

Not only did Catholic schools no longer have government funding, but the Catholic minority was on the whole less well-off than the Protestant majority. A big part of how Catholic schooling survived was the reliance on religious orders to provide teaching staff. However, by the 1960s, Australian Catholic schools were trying to deal with the post-war baby boom simultaneously with a crash in vocations to religious orders. Many Catholics were angry about the fact that their taxes paid for the education of Protestant children but not their own, while their own kids suffered from ballooning class sizes, overworked teachers and schools that were physically falling apart.

Much like in the US where Catholics traditionally supported the Democrats, Australian Catholics historically supported the centre-left Labor Party; the conservative side of politics was very Protestant. However, many Catholics were angry at the Labor Party for refusing Catholic demands for public funding for religious schools, despite relying on their vote to gain and remain in power; conservative Prime Minister Robert Menzies saw this as an opportunity to try to get Catholics to switch their political allegiance from the left to the right, so he went to the 1963 election with a policy of federal funding to construct science laboratories for all secondary schools without distinction (whether public or private, secular or religious), and he won, and the decision of many Catholics to switch sides played a big part in his victory. The Labor Party then realised that if they continued to oppose public funding of religious schools, they might lose the Catholic vote permanently, so they begrudgingly came around to supporting it too.