The functor of points approach (see around remark 2.3 in *geometry of physics -- supergeometry*) says that we understand a supermanifold \(X\) (or any other richer or more general kind of super-space) by remembering (in particular)

1. for each superpoint \(\mathbb{R}^{0\vert q}\)the set of maps \(Hom(\mathbb{R}^{0|q}, X)\) from \(\mathbb{R}^{0|q}\) to \(X\)

2. for each map of superpoints \(f : \mathbb{R}^{0|q} \to \mathbb{R}^{0|q'}\) the corresponding restriction map \(f^\ast : Hom(\mathbb{R}^{0|q'},X) \to Hom(\mathbb{R}^{0|q},X)\) given by precomposing any \(\mathbb{R}^{0|q} \to X\) with the given \(f\).

Now such maps of supermanifolds \(\mathbb{R}^{0|q} \to X\) are dually homomorphisms of their super-algebras of functions

\(C^\infty(X) \to C^\infty(\mathbb{R}^{0\vert q})\).

But the algebra of super-functions on the superpoint R^0|q, that is just the Grassmann algebra \(\wedge^\bullet \langle \theta_1, \cdots, \theta_q\rangle\) on q odd generators theta.

Hence such an algebra homomorphism is schematically of the form

\(f_0 \mapsto f_0(x) + \theta_1 \theta_2 f'_0(x) + \cdots\)

\(f_1 \mapsto \theta_1 f_1(x) + \theta_1 \theta_2 \theta_3 f'_1(x) + \cdots\)

where \(f_0\) is of even degree in \(C^\infty(X)\) and \(f_1\) of odd degree.

In other words, this is just the expansion of super-fields in terms of auxiliary Grassmann variables. This is indeed effectively the only way that supergeometry is presented in physics texts.

You see, the point is really to give a supply of Grassmann coordinates. In some texts on supergravity, there is a comment at the beginning saying something like "we fix once and for all an infinite-dimensional Grassmann algebra and assume that we may draw elements theta form it as need be". But there are pitfalls to this approach via "one single fixed infinite Grassmann algebra". These problems are discussed in

Christoph Sachse, "A Categorical Formulation of Superalgebra and Supergeometry" (arXiv:0802.4067)

The functor of points picture fixes this: instead of postulating one single infinite-Grassmann algebra, it says "your formulas need to make sense for Grassmann variables drawn from any finite Grassmann algebra" but (that's the functoriality) your formulas must be covariant under changing the chosen Grassmann algebra, i.e. under "Grassmann coordinate transformations".

In mathematical terms this: "work on all finite dimensional Grassmann algebras such that all your formulas are covariant under change of Grassmann coordinates", that just says that all the mathemaitcal objects you are dealing with are functors on the category of super-points.