The grocery chain campaign is well documented. Just search for it.
The short answer is that bags are a non-trivial cost for the larger chains. Now, they get to charge for them at an astounding markup and no longer have to compete with any grocery store on this point. All grocery stores are affected equally, which means it is disproportionately damaging to mom-and-pop stores and smaller chains.
How is this damaging to them at all? They literally get to cut one item completely off their expense list.
Grocery stores _absolutely_ supported the bag bans, though they weren't the initial groups asking for them. Similar to how the cigarette companies liked the TV ad bans--if nobody could advertise on TV than the playing field would be level and their profits all went up from decreased costs.
People here are talking about two kinds of laws: minimum bag charges and outright bag bans.
In some jurisdictions, a grocery store isn’t allowed to give you a traditional disposable bag at any price. In others, there’s either a bag tax or a minimum price, usually five or ten cents, a store must charge per bag.