You have a useful piece of kit there.
The black rectangular bar, (with the 1/4 or 5/16" spindle attached) can be fitted onto a Height Gauge, for use on a Surface Plate or table. It can also be clamped (carefully, and preferably protected on both dises) in a toolpost on the lathe.
The clamp will allow it to be fitted in a variety of places, but as has already been said, not to Surface plate or table.
The DTI can be used as a normal Plunger Clock, or as others have said, to check run out, (or position) of bores, using the pivoting accessory.
Quite versatile, you will find it very useful as time goes on.
Well Done for a good find!
Howard