The germ of a set in the origin $0 \in \mathbb C^n$ is given by a subset $X \subset \mathbb C^n$. Two subsets $X, Y \subset \mathbb C^n$ define the same germ if there exists an open neighborhood $0 \in U \subset \mathbb C^n$ with $U \cap X = U \cap Y$.
Given an element $ f \in \mathcal{O}_{\mathbb C^n, 0}$, $Z(f)$ is defined to be the set of zeros of $f$. $Z(f)$ is a germ by our definition.
Given a germ $X$, let $I(X)$ be the set of elements $ f \in \mathcal{O}_{\mathbb C^n, 0}$ such that $X \subset Z(f)$. But how do you make sense of the inclusion of germs?
For $f\in\mathcal O_{\mathbb C^{n},0}$,we denote by $Z(f)$ the germ of the zero set of $f$.Clearly, the germ $Z(f)$ does not depend on the chosen representative of $f$.
1.The inclusion of germs $A\subset B$ is defined by the condition that $\tilde{A}\subset\tilde{B}$ for some representatives $\tilde{A},\tilde{B}$ of those germs.
2.Thus,for sets $E,F\subset\mathbb {C}^n$,the inclusion $E_a\subset F_a$ means that $E\cap V\subset F\cap V$ for some neighbourhood $V$ of the point $a$，here，$E_a$ is the germ of a set in the point $a$.
3.I think in your statement,$X$ should be a germ in some point $a$,right? Then,$X\subset Z(f)$ means that $X\cap V\subset Z(f)\cap V$ for some neighbourhood $V$ of the point $a$.
4.For more details ,you can refer to the book $\textit Introduction\ to\ Complex \ Analytic \ Geometry$ by Stanislaw Lojasiewicz (page 80).