The obvious first pass solution here is that they distributed the meat to a lot of people. If everybody in a tribe gorges themselves on the meat from a hunt that can represent a lot more consumption than just an ordinary meal size.
There are a few additional food preservation techniques that we know were employed by paleolithic humans and a few that were likely as well but we have less evidence for. One is cold. Just as today, if you keep a piece of meat in a refrigerator it can last for a good long while before it becomes spoiled. Humans didn't have fridges back then but they did have other natural places where they could store food at chilly temperatures for extended periods away from land predators. One would be bodies of water such as lakes and streams. Especially during the ice age even in lowland areas many bodies of water would remain cool enough through many months of the year to provide some benefit to meat preservation. In the geographic areas where woolly mammoths were hunted typical lake temperatures are below 5 deg. C for as much as half the year or more, for example.
A similar technique would be to bury meat in the ground. Much like a body of water the ground temperature tends to remain closer to the average temperature throughout the day, and wouldn't be subjected to the higher peak daytime temperatures that would vastly accelerate spoilage. For many of the relevant geographic areas the average temperatures even today remain below 10 or even 5 deg. C for a substantial portion of the entire year, in the Pleistocene conditions would be even colder, providing a natural boost to meat preservation. Additionally, burying meat under the right conditions (which are very common and easy to achieve accidentally) can engage the process of fermentation which can protect the meat from spoilage that is harmful to humans. Indeed, this process of fermenting meat through burial is still a common cultural practice throughout the world, though particularly for high latitude or arctic cultures with a tradition of hunting (from Norway to Alaska).
Additionally, the times that humans and woolly mammoths coexisted were also when humans had mastery of fire, so it's quite possible that humans were preserving mammoth meat in the form of jerky and smoked meat even back then.
Edit: some additional reading: