If I had to make a guess, I'd guess there are two rubber rollers on either side of the core that grab it and push it down. When the core reaches the bottom of the slot, it likely pushes down on a circular plate. The under side of the circular plate is in contact with tabs on side panels, which have hinges just outside the edge of the circular plate. So the core pushes down on the plate, which in turn pushes down on the side panels, which closes the whole mechanism. There's probably some mechanism like the clicker on a pen to help hold the whole thing together when it's down too.
Here's the cut-away view of a quick mockup I put together in Fusion 360. The blue parts are the rubber rollers, the yellow are the plates around the edge, the purple at the bottom is the circular plate and the middle is your core. I didn't move the plates super precisely, but you get the idea.