It's an extension for those times that you need to drill really deep holes, or drill holes in hard-to-reach places, like through the fireblocking in a wall. It works with square taper shanks, like the square taper on the end of it. I'd have to have it in my hands to figure it out, but it probably starts with the two square openings aligned so you can insert the bit, followed by the squares getting out of alignment to trap the bit, and the inner square traveling somehow to tighten the bit up against the end square.
Start with lubricating it, then fool around to see if it becomes clear how it works. There were multiple different "chuck" designs on these extensions.
I see Geneg types faster than I do, but maybe the two of us together have provided useful information.