"Drawing a Picture" of $\operatorname{Spec}(\mathbb{Z})$

$(0)$ is the generic point. It is a single point, in some sense, "nowhere in particular" (quoting Vakil). A way to think of it is that closed points are actual points, and generic points are "fat/fuzzy" points (quoting here)

The generic point is drawn as going through all of the other points because the closure of the generic point is the entire space.