When performing principal components regression (PCR) on a high-dimensional dataset we are often looking for a few informative predictor variables in an ocean of uninformative variables. Principal components (linear combinations of the original explanatory variables) are used as regressors in our regression model. That way substantial dimensional reduction is often possible.
My understanding is that you still have to know the values of the original explanatory variables to calculate the principal components (since these are linear combinations of the original explanatory variables). Hence, we still need the same amount of values (of original explanatory variables) but the PCR regression model will include less terms.
$\bf My\ question\ is$: what exactly is the point of dimensional reduction? Clearly it is NOT to narrow down the number of values needed to make a prediction since that is unchanged.