I imagine it’s for setting up a specific machine for sizing some obscure thing. You set the width of the opening with the left side, height with the right. Now just start measuring random things, see what fits. Too small for olives and eggs.
We use a lot of screen cloth to size aggregates. The wire weaving companies give out for free simple guages to help measure screen openings and wire diameter. I usually just grab a tape. Here is one example, there is probably a similar application for your guage. Not sure how many I've lost over the years. Ed.
The tapered leaf is for measuring the distance between two wires in a woven mesh. ( think screen door screen only with 3/8" wire.) Like the starrett tool you linked, you slide it into a space till it stops and then read the width of that space. In this case the distance between the wires in the screen.