What was an insurance vending machine at an airport, and what kind of insurance did they sell?

"Was"? Travel insurance vending machines are alive and well in that wonderland of vending machines, Japan! Ryokouhoken-jidouhanbaiki (?????????) can be found to this day at most larger Japanese airports. For example, JII has machines at 8 airports, and here's a row at Kansai Int'l:

(courtesy this guy)

As for how they work etc, it's really no different to buying travel insurance online: you tell them how old you are, where you're going, for how long and whether you're planning to engage in mountain unicycling while sword-swallowing, and they spit out a quote based on your profile. Swipe your credit card or feed it good old cash, and ta-dah, you're set.

I remember that sometime in the 90ties I saw something like this at Charles de Gaulle Airport in Paris.

Of course different design - looked much more like a ticket vending machine of the time. But it sold insurances for a single flight.

Was even tempted to buy one as it was rather cheap.

Too bad I don't remember details, and no photo. But I think part of it was to give the address of the beneficiary, and that you would get an envelope to send a copy of the insurance policy to the beneficiary (in my case my wife). Somehow I thought this would be some kind of funny souvenir.

So it seems these machines still exist somewhere, or at least still did in the nineties.

I've changed this from a comment as it's longish and, while not strictly addressing the letter of the question, it does address the spirit, and adds some cautionary advice.

The equivalent functionality to an insurance selling machine is often achieved nowadays when an online ticket is bought as part of the overall transaction, using wholly customer-machine processes. The "machine" is now an internet connected computer or, really, the software running on a remote system. The insurance covers, at most, the flight being booked. It is usually life insurance with an allowance for baggage loss if this also occurs when you die. I have not seen broader 'travel insurance' offered in this way.

Unless the buyer's known travel-profile includes a known propensity for checked-in bombs, or carry-on knives and the like then their risk profile for the one flight involved should should be about the same as for all the rest on board.

Warning - Buyer Beware - "Insurance" may not be what it seems :-( :

Sometimes you may not only not get what you think you get, but effectively get a substantial negative return for your money. Always read the fine print when dealing with companies who you are not 101% confident in. (That should be most of them.) Because ...

We were offered such "insurance" on a flight from Yogyakata to Kuala Lumpur. The flight path was wholly in Indonesian territory except at the very end where it turned at 90 degrees opposite KL and flew into Malaysian air space.

I read the fine print. The policy excluded accidents in the airspace of the country that you took off from. That would mean it covered at best the maybe 5% of the flight which was in Malaysian airspace. Given that exclusion, and what I found next there was probably something there that excluded that leg as well.
I read on. By signing the document you assigned the "insurer" the rights to access all your medical information from any source in perpetuity. The right was non revocable by you or by your estate. This was specifically stated. As I recall the "insurance" cost about $US3. It seemed like a bargain. But, in fact, you were paying them $3 to acquire the rights to all your medical records irrevocably and forever. That right probably has substantial value in the insurance industry worldwide.

What was the airline? You may be able to work that out from the fact that one of their aircraft, piloted by an ex military pilot who was the son of a national leader, was lost with all aboard in uncertain circumstances (Airbus fall out of the sky mode) on a short international flight in the same general area that my flight was in, in the relatively recent past. I've wondered since what happened to passengers estates that tried to claim on the passengers insurance policies.