Rescaling of features

When we rescale our features, how can we use our model with real data, in other words
how we predict with parameters of our model with real data?
I find some results on the internet, however I’d like to understand the mathematical demonstration of those results, or at least the intuition behind them

When you rescale the features, you will get values for the mean and standard deviation.

Then you train and learn the weight values.

To make a prediction on new data, you apply the same scaling - subtract the mean anc divide by sigma. Then multiply by the weight and add the bias.