Even most grocery stores sell reusable bags for $1-2 apiece; so in the overwhelming majority of scenarios people would be paying more than 10 cents but less than the punitive $5. And by consciously buying a bag (instead of just saying "Yeah, I'll pay 10 cents for a bag" at the counter), people might be less likely to throw it out.
When that happens to me and you whack me for $5, I grimace a little, but basically don't care and my kids don't go hungry. If you whack someone for $5 who is poor and buying groceries with an EBT card, they're putting a gallon of milk back on the shelf to afford the bag. Or, they could ride the bus home, get their bags, ride the bus back, buy their stuff, and avoid the $4 fine ($1 for the bag and $4 for the convenience fine) at the cost of maybe an hour of their limited free time.
That's how it hurts the poor. That you or I could buy 10 bags for $12 and keep them in the trunk of our car so we always have them doesn't help the carless.
You asked how exactly it hurt the poor and I answered that question. I didn't come here spoiling for a fight over the issue.