The first is that the transmission channels for monetary policy are much wider than just the impacts on households with home loans. Most businesses have debts, too. 2 Floating rate debt costs more for them to service as interest rates rise, which presumably causes some of them to reconsider some things they might have been doing or planning.
So the “cash flow” channel of monetary policy affects business. Monetary policy also affects what economists would call inter temporal decision making. Incentives to save, as opposed to consume, alter. It is commonly observed that many Australians save rather little, but the household saving rate has in fact risen noticeably in recent years, largely unnoticed by conventional opinion.