The concept underlying renormalization/invariance as we now understand it, is a fundamental and profoundly beautiful idea about the very nature of what it means to make a model for a phenomenon. Essentially:
A theory is nothing but a way to turn (a finite number of) measurements into predictions.
The number of degrees of freedom in some putative model might be infinite, but all/most of them will be common between the observed context and the prediction context; it’s only the difference between those two that matters. And renormalization theory provides certain bookkeeping tools to track & ensure that, which then makes tractable the task of predicting.
A theory is nothing but a way to turn (a finite number of) measurements into predictions.
The number of degrees of freedom in some putative model might be infinite, but all/most of them will be common between the observed context and the prediction context; it’s only the difference between those two that matters. And renormalization theory provides certain bookkeeping tools to track & ensure that, which then makes tractable the task of predicting.