You should also have received a handleless stubby screwdriver as well as a full size jeweler's screwdriver. The part you're holding is part of the supplied tool to unscrew the bracelet mounting tube screws. The end of the stubby screwdriver fits into the hole of the piece in your pic and is held in place by the lock screw on the side. Use this device to hold the screw on one side of the tube in place while you unscrew the screw on the opposite side with the jeweler's screwdriver. You'll never need to use it unless you're changing to a bracelet, or the rubber strap wears out and needs replacement.
What Brent has said. I've just got one tiny thing to add - if ever you do change the strap be careful not to mark the watch. Oris give instructions how to use it with the Depth Gauge and suggest it is placed on a flat surface to engage the strap screw. By doing it this way you will have about as much control over it as I would have in a watch shop - not a lot. I use it, instead, as a normal screwdriver.
... it's also a good idea to put tape over the lugs for protection, then pierce the tape with the screwdriver to unscrew. It's very easy to scratch the lugs if either of the screwdrivers slips, and the tape will add some protection.
It's been my experience with both my Oris divers that they use a thread locking compound on the screws, and sometimes it leaks down between the tube and the lug making it very difficult to remove the screws and impossible to push the tube out to remove the strap/bracelet. Be very careful. If it becomes obvious that the screw and/or tube is locked tight by thread locking compound, then spot heat the end of the screw with a soldering iron to help release it. It's frustrating that they do that (overkill) and the problem is that they screw the tubes in while the thread locking compound is wet (a big 'no no'). If you want to lock the threads to prevent them from accidentally unscrewing, but make it possible to unscrew them in the future, then put a tiny bit of compound on the screw threads and let it dry first before screwing things together (never do it wet). Out of the four tubes in my two divers, I actually had to drill one out because it was completely locked in place (fortunately spares were provided with the watch).
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