I'm not sure if this is because I just woke up and my brain isn't awake yet, but how does this work? Where do you leave the grab bag and how do you receive one?

It's pretty simple really. Someone on the Dis decided there should be a grab bag type swap where Disers put their leftover stuff they would throw away that perhaps someone else could use. Items like ponchos, detergent, small kid's toys, crayons & such, non-perishable unopened food items, etc. Then the next person would use whatever out of the box they needed, and leave something else (to pay it forward.)
Swapping is easy. You look at the list to see who is after you (and before). If your dates overlap, you could try to arrange a face-to-face swap and hand off the box. If not, you exchange info (Full name, res #, check-in date, resort (Riverside or French Quarter) & cell # just in case). You would pick up the box at luggage services if you couldn't do a face-to-face. When checking out, you would put the next swapper's info on a card or piece of paper, attach it to the box & leave it w/ luggage services for the next person to pick up. That's the jist of it.
There's a Pay-it-foward swap that's been going on at the value resorts for a few months & it's been going great. You can check out one of their threads if you want to read about what they've been finding in their boxes.

I borrowed their idea after I had to drop out of their swap when I switched to POR.
Let us know if you want to be a part of our Pay It Forward Swap or if you have any other questions.

There's lots of room. The first page of this thread has the list of swappers & dates. It is up to date.