Here is why it is true:

1) Price inflation precedes wage inflation,

2) Low income households have very little cushion, they need every penny of income just to meet basic needs so,

3) When prices go up before wages, their very basic needs start to get cut, unlike in high-income households who easily cover their basic needs.

This is just obvious.

Does inflation shrink debt? Yes. People in debt have that debt shrunk in real term.

Does inflation shrink savings? Yes. People holding money lose value.

But neither of those affects day-to-day short-term living... paying the bills.