The people hurt in most tangible ways are the ones who don't have the capacity to save; if you were spending all your salary each month, living hand to mouth, inflation now means fewer and less quality goods and services. And that can be eating less nutritious food, having to use more worn down clothes, having to postpone medical/veterinary expenses further, etc.
Now, it happens that in the US the people working the least paid jobs have seen a wage increase, but that won't be necessarily the case and they aren't the only ones on a tight budget.