What is the intuition behind "states with support on orthogonal subspaces"?

Consider a vector space $V$ with an inner product and a linear operator $A:Vrightarrow V$.

The definition of support that you have posted from Wikipedia can be a bit confusing. It says $text{supp}(A) = {uin V|Auneq 0}$. This is the complement of the kernel where the kernel of $A$ is $text{ker}(A) = {vin V| Av = 0}$. 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. $text{supp}(A) = {uin V| langle u, vrangle = 0, vin text{ker}(A)}$. See for example page 14 of this textbook where it is introduced. Some useful properties are

1. For self-adjoint $A$, the image of $A$ is the same as the support of $A$. To see this, pick any $uin ker(A)$ and $vin V$. Then you have $$0 = langle Au, v rangle = langle u, Av rangle implies Av in text{supp}(A)$$
2. 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.