Starting from an informal understanding of blowing-up as replacing a subscheme by the possible directions into it (or some more accurate formulation of this), how does one justify the definition of the blow-up of $X$ along $Y$ by the formula
$ \mathrm{Bl}_Y(X) = \mathbf{Proj} \left( \mathrm{Rees}(\mathcal{I}) \right ) = \mathbf{Proj} \left ( \mathcal{O}_X \oplus \mathcal{I} \oplus \mathcal{I}^2 \oplus \cdots\right )?$
Here $\mathcal{I}$ is the ideal sheaf defining $Y$ in $X$ (with appropriate conditions as needed).
I'm not used to thinking about what $\mathbf{Proj}$ looks like; although things like $ \mathbf{Proj}\left ( \mathrm{Sym} \left ( (\mathcal{O}_S^{n+1})^\vee \right )\right ) = \mathbb{P}^n_S$ are obviously quite intuitive.
What happens if I consider instead $ \mathbf{Proj}\left ( \mathrm{Sym}(\mathcal{I} ) \right )$? Is this a related object, and what's its geometric interpretation? (In some simple examples these are closely related, as in the case of the blowup of the plane at the origin where $\mathrm{Sym}(\mathcal{I}) = \mathrm{Rees}(\mathcal{I})$; see comments to rattle's answer.)
(I'm aware of the universal property of blow-ups where this description fits in rather easily, although my understanding of $\mathbf{Proj}$ is sufficiently lacking for me to understand that properly; but that's not really the question I'm asking.)