Its for holding small cylindrical objects or, possibly threaded items like screws, in a vice so you can work on the end or head.
Usually threaded in appropriate sizes when used for screws or other threaded things to minimise risk of damage.
Basically slide or screw the part to be worked on in as appropriate and clamp up the vice. The slit lets it close up enough to grip hard but not so much as to damage the object,
Clive