The offset, log of population in your case, is just another predictor in the model. The only difference is that it is restricted to have a parameter estimate equal to 1. In the estimation process, all parameter estimates are adjusted for the presence of the others. Since the response function that you are modeling is the log of the Poisson mean, that is what the parameter estimates apply to. So, the X1 parameter is the effect of a unit increase in X1 on the log Poisson mean - the log mean count. However, computing a linear combination of the parameters, without involving the offset value, predicts the rate, not the mean. See the discussion of all this in this note.
BTW, you should never specify a variable in the CLASS statement that is not used elsewhere in the model specification. Missing values in such variables will cause observations to be ignored which don't need to be. Since you are not using COUNTY in the model and it is the only thing in the CLASS statement, you should drop the CLASS statement.
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