We punish because if we did not regard it as optimal to inflict disutility on those who game-theoretically defect, (a decision theory instantiated by) some people would not regard it as optimal to cooperate, and we would thus be in a worse position; and we cannot simultaneously be "such that we would punish", and reap all the benefits of subjunctive punishment, while not also punishing while in the branch that has a particular crime.

Note: this reason does not depend on any punishment, or even punishments in general, causing less disutility than not-punishing, in the technical sense of "cause" (which requires the effect to be either in the future, or, in timeless formalisms, a descendent of the minimal set [which includes the cause] in a Bayesian network that screens off knowledge about the effect). However, with an appropriate comparison to the counterfactual case, the punishers come out ahead.

(And yes, I'm neglecting the case of the exact limit of punishment for which this holds, and thus the issue of what it takes for a punishment to be regarded as excessive.)

The general pattern of "when people want to punish" coincides largely with what this reasoning gives, although internally it usually feels like some otherwise-ungrounded "deservedness of others of being treated badly" (see the Drescher quote in my article).