The trash seems to have been left in a reserved tent, so presumably they had a mailing address on file with the reservation.

They wrote down the ID during checkin. Standard procedure.

It's not an unanswered question. It's a meme that the government figured they could use to make people imagine the trash they leave being mailed to them by the government, which then implies the government knows where they live and who they are, whenever they leave trash.

Why does there have to be a means for the government to mail the trash back to the people, if just imagining the trash being mailed to them causes them to stop leaving trash?

I doubt you are going to leave trash, and the reason why is because you are asking hard questions of an illusion.

If you check in with your government issued ID then the government knows who you are and can find your registered address :)