Consider a vector space $V$ with an inner product and a linear operator $A:V\rightarrow V$.
The definition of support that you have posted from Wikipedia can be a bit confusing. It says $\text{supp}(A) = \{u\in V|Au\neq 0\}$. This is the complement of the kernel where the kernel of $A$ is $\text{ker}(A) = \{v\in V| Av = 0\}$. However, this definition of the support leaves you with a set that is not a vector space (for example, the zero vector is not in $\text{supp}(A)$).
The definition of the support of $A$ which is used in quantum information is the orthogonal complement of the kernel i.e. $\text{supp}(A) = \{u\in V| \langle u, v\rangle = 0, v\in \text{ker}(A)\}$. See for example page 14 of this textbook where it is introduced. Some useful properties are
- For self-adjoint $A$, the image of $A$ is the same as the support of $A$. To see this, pick any $u\in \ker(A)$ and $v\in V$. Then you have $$0 = \langle Au, v \rangle = \langle u, Av \rangle \implies Av \in \text{supp}(A)$$
- In finite dimensional space, taking the orthogonal complement twice brings you back to the original subspace. So the kernel is also the orthogonal complement of the support.
If $\rho$ and $\sigma$ have support on orthogonal subspaces, then it means that the $\text{supp}(\sigma)\subseteq \ker{(\rho)}$ and $\text{supp}(\rho)\subseteq \ker{(\sigma)}$. You can then construct a projective measurement $\{P_{\text{supp}(\rho)}, I - P_{\text{supp}(\rho)}\}$ or $\{P_{\text{supp}(\sigma)}, I - P_{\text{supp}(\sigma)}\}$ and this is guaranteed to distinguish perfectly between $\rho$ and $\sigma$.
TL;DR: The main reason why we care about states with support on orthogonal subspaces is because there exists a measurement that can distinguish between those states perfectly.
