Why do people bury chili underground?

Typically, dutch ovens and other heavy, especially cast iron pots would be buried to allow a slow dissipation of the heat. This will allow the contents to continue to cook at a very low and slow pace, usually to allow the release and melding of more flavor. In the case of chili for instance, this could allow more of a dried chili essence to reconstitute and become more uniform throughout the pot. The insulation of the ground will slow the cooling and allow this process to extend much longer than air would. Additionally, packing soil around the pot would tend to act as a barrier to hold in more of the aromatics than in air, combined with the slower cooling to again allow more absorption and a more even distribution for a more mellow taste to the dish.

Especially pots such as dutch ovens, this process is often done without additional heat, or in ground warmed by a fire, or with added coals or hot rocks depending on how long you want this process to continue and if you want more actual cooking to occur.

Fully duplicating this process inside would be difficult, but a somewhat close approximation would be to heat up an oven. Put in your covered pot, preferably tightly covered. Close the oven then turn it off or down to a very low setting, and keep it closed for a few hours and you should get much of the same effect. I would call this process marrying or melding the flavor rather than saving it.

This is, I'm pretty sure, a very old cooking method - and one with several modern kitchen equivalents. The earth (and charcoal) would trap heat, for slow cooking and remaining hot for an extended amount of time melding the flavors together. The longer it cooked, the better it would have tasted - especially for bean dishes, or stews (chili is kinda like both), which are known for improving by both longer coking times and extra resting times.

So, the very first ovens, earthen ovens, were not much like modern ovens - they were large earthen chambers, and a fire was built inside them to heat the earthworks - and when it was hot enough, the fire was doused and raked out and the food placed in the preheated oven to cook in the residual heat (thickly walled as they were, and as hot as they got, an oven would still hold heat for hours and hours). They were actually not heated directly, it was all preheating. This gave rise to some types of dishes, baked beans is one example, that were cooked for extended periods on inevitably diminishing heat - sometimes as long as overnight - and which were improved in flavor for having the extra time cooking.

From there, and kinda hand-waving lots of experimentation and variations, there are a lot of variations for how the technique evolved and changed. The appliance to replica the strategy in a modern kitchen, will depend on which aspect the recipe called for. The modern oven is one adaption, with a dry heat and higher temperatures, another is the slow cooker with lower but more consistent temps and better moisture control - both holding at consistent-ish temperatures (trading the continual heating for the extra insulation to store that thermal energy). The fridge is usually used for extended resting times (stews and bean dishes are often better the next day) - the tastes might not meld as much at fridge-temps, but it's safer to hold the food that way, and even fridge-resting improves the flavor.

As for the actual technique you saw, the modern dutch ovens used the same sort of process - preheating the pot on the fire, adding the food, and partly burying it in the ashes to cook. Burying it in hot ashes, or with heated coals, would create a sort of slow-cooker method (where the food was placed in the heated environment and sealed (as much as possible) to trap the heat. Burying in earth (in the literal sense) seems like it might be from a camping technique, which can be a form of slow cooking (with coals), or a form of better insulating the pot (to cook with residual heat for longer times, re-inventing the oven), or even a preservation method - from wandering tasters, er, I mean bears, or possibly even from spoilage.

Yeah, another process which might be relevant to the reasons for doing so might be canning - if the food is heated to safe temperatures (which should kill the bacteria), and can be stored for a while as long as it isn't exposed to air or new bacteria - say, being cooked hot enough, then left buried underground, or sealed in an earthen oven (practically-ish sealed form the open air and random organisms therein between lid, earth, and heat-scorched air), from the time it cooled until it was opened up and used - then it might be safe to leave still hot and allowed to cool overnight, or perhaps not dug up till days later.

In more modern times, the most direct equivalent to that specific technique would probably be a warming drawer - the heat and rest time after the cooking process is complete lets the flavors meld together much more quickly than, say, being refrigerated - while the food was being kept safe , or nearly so, at calculated temps. Of course, modifying the chili recipe to use an oven or slow cooker to being with might create very similar effects when originally cooking the dish, long and slow cooking will let the flavors meld rather more than cooking on the stove or over a fire (given these contestants were in a place to bury the chili to begin with). Or even cooked normally, then placed in a slow cooker or oven to finish a long, low heat melding process. There may be subtle specific textures or flavors brought about by the high temp cooking process owed by a slow cooling process (which modern cooking isn't really equipped to do, it's all constant consistent heat till its turned off) - but I expect a version slow-cooked in an oven or crockpot to be very similar to the earth-buried version.