# Pearson's R as a measure of noise between two linearly related variables

I know that the Pearson's correlation co-efficient the Pearson's correlation co-efficient $$R$$ is defined as: $$R=\frac{\mathrm{Cov}(X,Y)}{S_X S_Y}$$ and that $$-1 \leq R \leq +1$$ as a result of the Schwarz inequality, but I can get no intuition whatsoever as to why $$R^2$$ is a good statistic representing the noisy nature of the linear dependence of $$X$$ and $$Y$$. Is there a more rigorous derivation which defines it from variance of the error signal?

I mean if I take a dataset $$(X,Y)$$ and fit a line $$\hat{Y}= aX +b$$ using OLS to it. Shouldn't the variance of the random variable $$Z=Y-\hat{Y}$$ itself be the best estimate as to the goodness of the fit/"noisy nature of linear dependence", why does $$R^2$$ come into the picture here?