Let $X$ be a scheme. Choose $C\subset X$ be a subscheme of $X$ and let $\mathcal{I}\subset \mathcal{O}$ be the corresponding ideal sheaf. Then $\mathcal{B}=\oplus_{d\ge0}\mathcal{I}^d$ is a sheaf of $\mathcal{O}$-algebra The blow-up of $X$ along $C$ is defined as $ Y=Proj \mathcal{B} \rightarrow X. $ My question is, how can one understand $Proj \mathcal{B}$ to $see$ geometric description of blow-up? More precisely, when both $X$ and $C$ are smooth complex variety, $Y$ is obtained by replacing $C$ by $\mathbb{P}(N_{C/X})$, but I cann't really see this description from $Proj \mathcal{B}$.
THank you for your help.