The tool in the listing is incomplete; there is an extension that screws into the rod that is there.
As woodchucker says, this is a tool for centering work in a 4 jaw chuck that may be difficult to center. A square or rectangular work piece comes to mind. You cannot indicate off the surface of the work so you use a tool like this one. A center punch marks your desired center on the work. The pointed tip of the long rod is extended about an inch or so and locked into the gimbal before being placed into the punch mark. Then you rotate the chuck by hand and watch the other end of the long rod; it will swing in a large circle initially but as you adjust the jaws on the chuck the circle gets smaller until you get the part centered. At that point, the end of the rod is barely moving, if at all. This is why the extension for the rod is important; the longer the rod, the more accurately you can center the work.
There are other ways to do this nowadays but back when this tool was made it was a pretty useful idea. That part that holds the rod is a gimbal that can rotate 360 degrees in all directions and the tool is very nicely made. I have a complete tool and while it is rarely used, it is a beautiful example of an old Starrett tool.
Edit: I know Brown & Sharpe also made one and I think Lufkin did, too. I forgot to add that the big rectangle thing goes in the tool holder; obvious but I wanted to be complete.