- retribution: gives victims and families of the victim retribution. This one doesn't seem important in the modern day, but if you ignore it you get vigilante revenge, blood feuds, honor culture etc. Having the state take on "revenge" kills these cycles.
- incapacitance: straight up preventing dangerous people from being out and being dangerous.
- deterrence: long prison sentences deter other people from committing crimes because they don't want to get a long prison sentence. Also deters the prisoners themselves from committing another crime (if their sentence is short enough)
- rehabilitation: here's finally the "corrections" part, and honestly our system in the US isn't great at rehabilitating anyone. In principle this is a prudent idea, but we'd have to spend a lot more tax dollars on prisoners to actually rehabilitate anything. That's politically pretty unappealing