Documenting a language is how you "save" it. When a language is dying out, that means people are no longer learning it, which implies that native speakers are not using it to communicate with their children. One reason for people to use a different language with their children is to give them a headstart in school, where another language is used for instruction, and to avoid the stigma that comes with speaking a less prestigious language. If their own language is closely related to the prestige language of the region (very likely in case of a contact language) it's easy to compare the two, notice that they're different and conclude that the prestige language is "correct" while anything different from it is "wrong". That can lead to extreme social pressure. If you've ever made an embarrassing mispronunciation, you'll know the feeling, but now imagine everything you say is "incorrect".

Language documentation doesn't completely change the social environment, but can indirectly enhance the language's prestige, e.g. by devising a writing system that then gets adopted in education and media and by raising awareness that "different" doesn't necessarily mean "wrong". It also means that children who grew up without learning their parent's mother tongues but want to learn them later in life can use the material collected as part of language documentation for studying.