Why does a Weil Divisor $ D $ on a scheme $ X $ induce a Weil Divisor $ D_{x} $ on the local scheme $ \text{Spec}(\mathcal{O}_{X,x})$?

Question

Let $ X $ be an integral, separated noetherian scheme, and $ x \in X. $

In Proposition 6.11 of Hartshorne's Algebraic Geometry(Page 141), the author states that if $ D $ is a Weil divisor on $ X, $ then $ D $ induces a Weil divisor $ D_{x} $ on $ \text{Spec}(\mathcal{O}_{X,x}). $

I am not seeing why this follows.