What are the ways punishment reduces crime?

Through negative reinforcement. The perception is, is if the punishment is severe enough and they think they'll get caught, they'll want to avoid the punishment by not doing the crime. The same function you discipline a child for being bad.
It doesn't reduce crime. If imprisoned I guess it reduce crime because the opportunity to commit a crime is not there. But other than that, it is a risk reward situation and someone will always be predisposed to take the risk. So there is no effective deterrent.
Take Saudi Arabia for example. The most unacceptable deterrent thay have is the capital punishment of beheading in a public square. If it be an effective deterrent there would have be only one execution. Over the centuries there have be thousands upon thousands. People take the risk and loose their heads. In the US it is the death cost. Still doesn't work. In theory, a time of incarceration is supposed to give the criminal time to think give or take a few their deeds and to reflect on how they may do better in the future.

There are lots services and programs offered by the prision systems. In the past seven or so years, most have become "faith based". There is nil yet to document the results of those particular programs.

"Punishment" as such has no history of solving criminal behavior. In argument a released convict is a clean slate with an opportunity for a different future. In truth, a narrative of a conviction and incarceration is a form of punishment that takes considerable energy and resources to overcome after release.

There is considerable documentation that illustrates that by and roomy, released prisoners are more capable of restarting a criminal life with advanced fluency and capabilities learned in prison.

Punishment reduce crime if that punishment is based on education, skills training and eventual employment.
Offenders are supposed to learn not to repeat the crime and they're supposed to set an example for others not to commit the crime.

The Muse