In summer, the usual position of the belt of westerly winds is further south of Australia compared to winter. Rainfall from cold fronts and troughs is far less common in summertime, while the east gets more rain coming in from waters to the east of the country.
In a positive SAM phase, the belt of westerly winds contracts towards Antarctica. Since the westerly winds and high pressure are already further south below the continent, the southward movement only acts to decrease rain events for western Tasmania. In eastern Australia, the southward movement of the westerly winds means that more moist onshore flow from the Tasman and Coral seas is drawn inland, and thus increases rainfall for eastern Australia. This effect for eastern parts of Australia is much more widespread in summer as the east coast climatologically receives its highest rainfall in the summertime.
In a negative SAM phase, the belt of westerly winds expands towards the equator and Australia. Shifting the westerly winds to the north in summer means less moist onshore flow from the east, and thus typically decreases rainfall over eastern Australia. For Tasmania, the northward displacement of the westerlies means a stronger than normal westerly flow over the island, enhancing rainfall in the west.