What is the geometric significance of the definition of supermanifold?

Question

We know that a supermanifold $M$ is a locally ringed space $(M,O_M)$ which is locally isomorphic to $(U, C^\infty(U) \otimes \wedge W^\ast)$, where $U$ is an open subset of $\mathbb{R}^n$, $W$ is a finite dimensional real vector space and the above isomorphism defined in the category of $\mathbb{Z}_2$ graded algebra i.e. the parity $\bigoplus_{k \geq 0} C^\infty(U) \otimes \wedge^k W^\ast \rightarrow \mathbb{Z}_2$ defined by $ f \otimes x \rightarrow |f \otimes x| := |x| = k \mod 2$. I would like to know how can we geometrically think of this. For example we know that the algebra of differential forms $\Omega(M)$ on a manifold $M$ which is locally isomorphic to $C^\infty(U) \otimes T_x ^\ast M$ for some $x \in U$, therefore the sheaf of differential forms on a manifold corresponds to a supermanifold. How can we geometrically visualize this? Moreover what is the significance of defining a supermanifold structure for the sheaf of differential forms for a manifold $M$.