falling household savings
If household debt rises, it means that more households consume more than they earn. This is the same as saying they save less. This means that the causal channel in your quote is by definition. 

Now one may ask why do savings decrease in a period of growth. I could imagine two causal channels. It could be that growth is a consequence of low interest rates, causing both higher private debt and hence more investment. One could also think of sustainable growth causing positive expectations, so households are worrying less about the future and hence save less for precautionary reasons.