It's actually a rather simple effect that's carried off very, very well.
First, the thud and boom you hear and feel are effects. There's a bass floor-shaker installed in that section. (Note that it's the only area of the post-show where the floor is raised slightly)
Also, the roll of sheet metal at the front is separate from the rest of the effect, and just moves at the proper time to aid in the effect.
The chain of objects that move underneath the press is made up of only three pieces: one flat piece of metal, and two already-formed car parts.
What you normally see when the press rises up, is that the whole chain moves away from you, advancing the two car parts into the hole in the wall, and moving the flat piece of metal into the press.
Then the press head slams down.
What you don't see is that the moving head of the press is hollow. It has an opening at the back, facing away from you to allow the formed part to move while it's down. So while the press is in the down position and hiding the movement, the entire chain shifts backwards one space, bringing the part that just moved away from the press back inside the press, and the flat piece of metal underneath the front part of the machine.
So when the press head rises up again, what was a flat piece of metal has now become the "pressed" car part.
Because you don't see the chain moving in any direction other than away from you, you don't think that it would go backwards in that moment that you can't see it. It just cycles this way over and over. Press up, chain forward, press down, chain back. Up, forward, down, back.
If you get down on your knee so that the press is at eye level and look carefully, you can sometimes glimpse the chain of pieces just start to move backwards as the press slams down.
Sometimes the simplest tricks are the most effective when used in the right way (just look at the Haunted Mansion Ballroom) .
-Rob