Let us have $X$ a projective, smooth Calabi-Yau 3-fold and $Z \subset X$ be a subscheme supported on curves and points. Its structure sheaf $\mathcal{O}_{Z}$ fits into the short exact sequence
$$0 \to \mathcal{I}_{Z} \to \mathcal{O}_{X} \to \mathcal{O}_{Z} \to 0.$$
where $\mathcal{I}_Z$ is the ideal sheaf (supported on $Z$?) This is a generic statement but I do not quite understand it, or the information I can get from this.
What I understand, more diff. geometrically, is the following: On $X$ we have the ring of regular functions $\mathcal{O}_X$ and thus we have a corresponding sheaf of $\mathcal{O}_X$-modules. There exist submanifolds (subschemes in the language above) $Z$ whose ring of regular functions are $\mathcal{O}_Z$ which can be embedded (included?) in $X$ (I guess) as $$ \mathcal{O}_Z \hookrightarrow \mathcal{O}_X $$ So $Z$ could be 2 dimensional or 1 dimensional 0r 0 dimensional. Then over these submanifolds we have their local ring of functions forming a subring of the ring of functions of the whole space. What is the role of the ideal sheaf $\mathcal{I}_Z$ then?
What I ask is a more differential geometric way to understand ideal sheaves supported on closed subschemes. If there is a vector bundle version of this story that would be even better (i.e. focus on locally free sheaves).
Any help with this intuition or reference would be great.