On the informal interpretation of the scheme-theoretic image

Question

I am having some difficulties understanding the comment in Vakil's FOAG (Jul31, 2023 version) on the definition of the scheme-theoretic image. Let me write down the definition he uses:
Definition. Suppose $i:Z\hookrightarrow Y$ is a closed subscheme, giving an exact sequence $$0\to \mathscr{I}_{Z/Y}\to \mathscr{O}_Y\to i_*\mathscr{O}_Z\to 0 $$ we say that the image of $\pi:X\to Y$ lies in $Z$ if the composition $$\mathscr{I}_{Z/Y}\to \mathscr{O}_Y\to \pi_*\mathscr{O}_X$$ is zero.
Then there is the mysterious comment I do not understand:
"Informally, locally, functions vanishing on $Z$ pull back to the zero functions on $X$"
I understand when one says that a function (section) vanishes at some point of a scheme. But I do not have any idea about what this comment means here. Any help is appreciated!