Just like all Australia is like Queensland.
This is the story in question regarding Tegan Leach who is currently charged with procuring drugs for an abortion.
This article from the Courier Mail states that the laws that she is being charged with are actually over a century old.
Now the issue isn't "Why does Australia have such crappy abortion laws" but it's in fact "Why does Queensland have such crappy abortion laws". The issue of abortion is not a Federal issue, but a state one. In general whatever isn't explicitly stated in the Constitution as something the Federal Government is in charge of is the job of the States to deal with. That is why Marriage law is dealt with on a Federal level, but de facto relationships are dealt with on a State level.
Now to be fair, the state I live in (South Australia) actually has some rather crappy laws regarding abortion compared to other parts of Australia. Wikipedia has a nice picture summing it up here.
Queensland does not even allow abortion in rape cases.
Victoria and the ACT allow abortion on request (there might be limits I don't know about though...) and they are basically the polar opposite of Queensland in this sense.
With time, and also from the reactions that people have against those who firebomb the houses of people charged with violating the abortion laws, the view should change and Queensland should start to come in line with the rest of the country.