Translation:I come from Australia.
出身 refers to an origin where one is born, raised or educated. The origin can be a place, an institution, a social class, etc. I am from Australia / I come from Australia probably can deliver the same meaning.
I'm confused. In this same lesson, there was a question that had どの国のしゅっしんですか in it, where the object and しゅっしん were connected with の. Is that not always necessary?
As I understand, when 2 nouns go frequently together, they become close to a compound noun and の becomes unnecessary. However in どの国のしゅっしん, どの国 is not a defined term and I think because of this it cannot have a compound noun property. の is therefore still needed.
It is usually implied, but not always the case. For example, I wasn't born in Australia, but I moved there with my family when I was 5. I consider myself Australian, and I would say I'm オーストラリア出身, but I'm actually クック諸島産まれ (kukku shotou umare = "born in the Cook Islands")