... that allows the steps to collapse and the support brace to pivot inwards so that the unit can lay flat? Some sort of bolt?
This could be done multiple ways but I'd say there's a good chance the tread and the riser pivot on dowels. This is the simplest way to make anything like this where a component needs to rotate about an axis, although you could use steel pins* for increased strength and durability a wood-only solution can be strong enough and durable enough for a long service life. The key thing then would be the diameter and strength of the wood used for the dowel.
*Housed directly in wood or in bushings of plastic or metal