This question comes from Example 12.21 (a) of Gathmann's 2019 notes, here.
In the example, take $R = K[x]/(x^2)$ for some field $K$, so that $\operatorname{Spec} R$ is a single point $\mathfrak{p} = (x)$. In the third paragraph of the example, he says
Geometrically, one can think of $\operatorname{Spec} R$ as “a point that extends infinitesimally in one direction”: As on the affine line $\mathbb{A}^1_K$, there are polynomial functions in one variable on $\operatorname{Spec} R$, but the space is such an infinitesimally small neighborhood of the origin that we can only see the linearization of the functions on it, and that it does not contain any actual points except 0.
I am a little confused by this wording, so any clarification on what he means, or otherwise how else to visualise $\operatorname{Spec} R$ would be appreciated.
I have tried to reason it as follows. The space $R$ can be thought of as a two-dimensional $K$-vector space, one basis element given by $1$ and another by $x$. Any $f \in R$ takes the form $f = a+bx$, so the distinguished opens $D(f)$ are either $\operatorname{Spec} R$ itself for $a \neq 0$, or the empty set for $a = 0$. In this way, the single point of $\operatorname{Spec} R$ lies only 'over' the entire 'constant axis' of $R$ minus the origin, which can 'collapse' into a fat point at the origin. However I'm not sure if I'm correct here - what does he mean by "the space" (bolded)? $\operatorname{Spec} R$ or $\mathbb{A}^1_K$ or something else?