You should expect the RMSE to be the same between the two models. Standardization simply changes the units of the predictors, not their predictive power. Modeling an outcome using a predictor in inches will provide the same degree of prediction as modeling the outcome using a predictor in cm.
The coefficients in the unstandardized model can be interpreted as the expected change in the log of the odds of the response occurring corresponding to a 1-unit increase in the predictor holding constant all other predictors in the model. Log odds ratios are typically challenging to interpret, so sometimes people exponentiate the coefficients, which yields odds ratios. Odds ratios are also challenging to interpret; to get probability differences, you need to use a marginal effects procedure, which is available in the
Another interpretation is the "latent variable" interpretation. Imagine a continuous latent variable that is categorized either as 0 or 1 based on whether the value of that variable passes some threshold. Logistic regression can be seen as estimating a linear model for the distribution of the latent variable and the threshold. In this way, the coefficients represent the effects of the predictors on the latent variable. This latent variable is not easily interpretable but vaguely represents the propensity for the response to occur.
The standardized coefficients have the same interpretation, except that the units of the predictors are now standard deviation units. Thus, the interpretation of each coefficient is the expected change in the log odds of the response corresponding to a 1-standard deviation in the predictor, holding constant the other predictors in the model.