I am going to guess, so don’t laugh at me.
-If you've never had the need for one then you're simply not familiar with what it is. Anybody that would laugh at you for that is an idiot.
It is something that was very very expensive.
-They were (pre 1980's) because it was the only affordable means of obtaining accurate 4 place (.xxxx) dimensions larger than a stack of gauge blocks and easier to use than a micrometer for accurate numbers. They were used with a surface plate (for "zero" reference) and an indicator. There were a few different makers/models with different features (and priced accordingly) and different heights. The earlier models were like the first two shown (Cadillac PLA-CHEK) that had a shaft with only a single surface (top of disc) for measurement. A small "horseshoe" device was available to clip on when needing to reference a surface underneath instead of on top. A later design (Mitutoyo was first that I recall) used a different (and more expensive) version similar to the Brown & Sharpe model
@Firebrick43 posted above. This version has two surfaces for each 1" range to use for over/under and requires no "horseshoe" attachment for an underneath surface reading. The single shaft design became a dated but less expensive option.
And CNC has made it mostly obsolete.
-It's a dated design but I wouldn't call it obsolete and certainly not by CNC. The electronic models may have replaced (new toy syndrome) the mechanical type in the inspection dept. (data logging) but many people still use/prefer the mechanical design. The CMM (coordinate measuring machine) also drew favor with QC people but the simplicity, cost, and footprint of the mechanical types is still favored by many. I have a couple of different height models (smaller is easier to use) and prefer them over an electronic model or a CMM. What others prefer largely depends upon what you need it for and how accurate you need to be. All three have advantages and disadvantages.
-As close as a non-shop person (civilian) would get, you did ok. Asking questions is how we all learn.
What does the label say (besides Starrett)?
-There's a few other makers besides Starrett, they may have discontinued theirs. Sometimes called a Height Master but that name will usually bring up the Mitutoyo model.