Let consider $\mathbb{A}^2$ as general affine scheme. Which meaning do have "points" of the shape $(x,y) \in \mathbb{A}^2$? For example the special case $(0,0)$?
That's almost clear that for choosen (therefore fixed) field $k$ we can identify the points $(x,y)$ for $x, y \in k$ with $k$-points $Spec(k) = {*} \to \mathbb{A}^2$. Concretly: $(x,y)\cong (t_1-x, t_2-y)$ is the maximal ideal (thererfore element of Spec) with indeterminats $t_1, t_2$.
But the meaning of this "points" of the affine scheme $\mathbb{A}^2$ without mentioning a fixed field isn't clear to me.