They used boats.
The sea level was lower then, and many of what are now islands were then part of the mainland. For example, one could walk from present-day mainland SE Asia to many of the Indonesia islands, and Australia and New Guinea were joined together. Still, there were significant water barriers.
The two most likely routes that were used were (a) along Java through to Timor, at which point there was an 80-90km water crossing to reach Australia and (b) through Borneo to Sulawesi, at which point there was a slightly smaller water crossing to reach western New Guinea. Both gaps were far to wide to feasibly swim; boats were used. This wasn't just going out into the unknown - both gaps should have been crossable without ever going out of sight of land, and offshore Australian islands may well have been visible from Timor.
While this was a lot longer ago than the age of the oldest recovered boat, this isn't that surprising, since wood survives poorly in the archaeological record. For example, the oldest boomerangs found in Australia are about 10,000 years old, while rock art conservatively dated to about 20,000 years ago (and less conservatively up to 50,000 years ago) shows boomerangs. The oldest boomerang ever found is about 23,000 years old - but made of mammoth ivory rather than wood (from Poland). Boats are much, much older than the oldest surviving identifiable remains of a boat. (Some Australian rock art showing boats has been dated to about 17,000 years ago. This date is not generally accepted.)
More generally, the Stone Age isn't the Stone Age because most tools back then were made of stone - most of their tools would have been wood, bone, antler, hide, horn, etc. It's just that those materials haven't survived well, while stone can last millions of years.
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