The claim, although surprising, is easy enough to verify with a quick search online. According to Geoscience Australia, an agency of the Australian Government that carries out geoscientific research, Australia is approximately 3860 kilometres long from its most northerly point to its most southerly point in Tasmania, and is almost 4000 kilometres wide, from east to west.
In contrast, NASA states that the diameter of the moon is approximately 3475 kilometres, or 525km less wide than Australia. The claim that Australia is wider than the moon is therefore true.
What the claim did not consider however was surface area, with Australia being “flat” while the moon is a spherical shape. Comparing surface areas, the moon is far bigger; the surface area of Australia is approximately 7,617,930 square kilometres, while the moon’s surface area is around 37,940,000 sq km, making it almost five times the size of Australia.