An Introduction to Statistical Learning puts the rationale pretty succinctly on page 214:
The rationale here is that if a set of models appear to be more or less equally good, then we might as well choose the simplest model—that is, the model with the smallest number of predictors.
Note that the one-standard-error rule is recommended in cases for which the relation of cross-validated error to number of predictors is "quite flat." It also takes into account the following:
Furthermore, if we repeated the validation set approach using a different split of the data into a training set and a validation set, or if we repeated cross-validation using a different set of cross-validation folds, then the precise model with the lowest estimated test error would surely change.
So why not choose the simplest useful model in such a case? The one-standard-error rule is a rule of thumb, a way to get a reasonably simple model. I am not aware of any further basis for this choice based on first principles.
This Cross Validated question goes into a good deal more detail on this issue.