There’s no reason to suppose that it’s connected to the Australian gun laws, unless the burglary rate is shown to have climbed since the more restrictive gun laws were introduced. I couldn’t get nationmaster.com to generate that kind of information.
My WAG is that (legitimate) gun ownership doesn’t stimulate crime and restricting gun ownership doesn’t prevent (very much) crime, but there may nevertheless be a correlation, in that social attitudes to gun ownership and social attitudes to crime may be affected by common factors. E.g. societies which are relativly tolerant of the use of force are likely to be relatively tolerant of/open to gun ownership and to certain kinds of crime.
Why does Australia have a high burglary rate? I don’t know, but I doubt very much if it is due to restrictions on gun ownership. We cannot assert that it is unless we are also prepared to explain why (say) Ireland, a country with per capita income not dissimilar to the US or Australia, whose gun laws are even more restrictive than Australia’s, has a lower burglary rate than the US. (It’s no. 19 in the list.)
My WAG (again) is that the Australian statistic is explained at least partly by the fact that Australia is one of the world’s most urbanised countries, and burglary is predominantly an urban crime. You expect higher burgary rates in more urbanised countries. But I’m sure there are lots of other factors at work as well.