How ancient peoples handled the meat from large animals varied over time, of course. Very early on it was a simple matter, a tribe that had a hunting party that took down a large animal (such as a mammoth or elk) would butcher it and bring the meat back which the tribe would consume essentially as a single meal. It's possible that they would even share a carcass with nearby tribes, something that many more recent hunter-gatherers do, given that successful kills would be rare and it would benefit everyone to ensure that any one success benefited (kept alive) the largest number of people.

Over time as humans became more skillful and successful at hunting and as people learned various tricks and techniques of food preservation they became able to save food over longer periods of time and did not have to consume the entirety of a single kill in one sitting. One of the oldest and simplest techniques of preservation is smoking. Simply cutting meat into thin strips and leaving it in close proximity to a fire will result in producing a smoked meat product that has a shelf life not just of days but of months. A similar technique is used to dry lean meat in the sun, producing jerky, also with a very long shelf-life. Jerky can be combined with dried fruits and fat to be used as a sort of "whole meal" bar or an emergency ration with a very long shelf-life, a version of which called "Wasna" continues to be made by the Lakota today.

Meat can also be preserved in lakes or in shallow underground pits. In the right climates this can preserve the meat in freezing or near freezing conditions. However, more importantly, in more wide ranging climates it can allow natural processes of fermentation to occur, which under the right (but not exceptional) circumstances can leave the meat in an edible (if sometimes questionably appetizing) state. There's some archaeological evidence of this along with tremendous anthropological evidence (including among more recent hunter-gatherer cultures) of this being a fairly common practice.

Edit: I should note that this and similar practices continues to be practiced today even in the industrialized world (just as making smoked meats and jerky is). Dry aging meat is yet another important food preservation technique that requires very little equipment and primarily just patience. Whereas meat fermentation encourages the growth of "friendly" bacteria which help protect against colonization by more harmful to humans microorganisms (such as botulinum) dry aging encourages the growth of a "crust" of mold on the meat which serves a similar role. Additionally, the dehydration of the meat makes it a much less inviting environment for bacterial colonization while still preserving the calorie content. As an added benefit aging allows for the action of natural enzymes to soften the meat and produce different, usually desirable flavors. This has, of course, become a tradition carried forward through the millenia in the form of making hams, bacon, and so forth.

What's important about what I listed above is how readily available these techniques are to any sort of paleolithic humans. If you have knives, sticks, and fire, you can do any of it. Moreover, these techniques aren't exactly rocket science. Yes, there's an art to optimizing them, but each of them can be easily discovered accidentally through doing ordinary things with meat, and once discovered they can be fine tuned through trial and error and made into a common practice. If you are cooking your meat you will almost inevitably accidentally smoke some of it. And if you figure out how to min/max smoking you'll naturally fall into making jerky. It could be as easy as preparing thin strips of meat for smoking during a hot summer day, laying them out in preparation for bringing them to a fire and then discovering that leaving them in the sun over just half a day turned them into something different. Similarly, hiding meat from predators in water or underground and then returning to it later than expected can lead to the accidental discovery of fermentation. As they say "if it's stupid and it works it's not stupid".

Salt has also been used since prehistoric times for meat preservation but, of course, it was usually not available everywhere in abundance, so it was far from universal.

Meat was hardly the only component of the diet of hunter-gatherers however, and many of the other foods (such as nuts and seeds) had a longer shelf life than unprocessed meat. As the paleolithic transitioned to the neolithic our ancestors increased their skill at preserving calories over long periods by leaps and bounds. They began cultivating grains that produced highly durable and long-lived kernels. You take a large amount of wheat or barley kernels and you place it in a dry, covered hole in the ground and the kernels around the outside begin sprouting just a little, which uses up oxygen and fills the space with CO2, providing an excellent environment for preserving the bulk of the kernels for periods up to even years. They began keeping some game in controlled herds, eventually resulting in domestication (what better way to preserve meat than on the hoof?) They began to more effectively process foods into secondary products (not just the Wasna or Pemmican as above) with things like olive oil, rendered fat (tallow, suet, etc.), dairy (milk, cheese), wine, mead, beer, breads, etc. It's this combination of technologies which dramatically improved the diets of humans through the lean winter months and in particular dramatically lowered infant mortality from their hunter-gatherer levels, sparking the boom that would eventually result in the growth of the worldwide human population in the holocene to billions over a period that was just a fraction of the time span of even the upper or late paleolithic period before it. There is some evidence that human populations began growing substantially near the end of the paleolithic even before the proper "neolithic revolution" hit full swing. Whether this was due to better hunting techniques, better food preservation skills, or other factors entirely is still an open question.