= Solution
The microscopic electric interaction is <spin> independent to leading order, but the <Pauli exclusion principle> ties the spatial symmetry of an electronic state to its <spin> symmetry. Different <spin> combinations therefore have different Coulomb and kinetic energies. Projecting those energies onto localized low-energy moments often gives an approximately rotationally invariant bilinear exchange <Hamiltonian operator> $-J\mathbf S_1\cdot\mathbf S_2$. For <spin> one half, the <spin singlet state> and <triplet state> energies are distinguished by $\mathbf S_1\cdot\mathbf S_2=-3/4$ and $1/4$, so any isotropic two-level <spin singlet state>–<triplet state> splitting can be written in this form plus a constant. More general <spins> or orbital degeneracy can require extra interactions.
An antiferromagnetic oxide commonly has <superexchange> through a ligand such as oxygen. Virtual hopping between half-filled metal orbitals lowers the energy of antiparallel <spins> more strongly because the corresponding intermediate doubly occupied configurations are allowed by the <Pauli exclusion principle>. The simple one-orbital <Hubbard model> limit gives $+4t^2\mathbf S_1\cdot\mathbf S_2/U$, equivalent to negative $J$ in the question's minus-sign convention. Bond angles and orbital occupancy can change the sign in more complicated oxides, so <superexchange> is not universally antiferromagnetic irrespective of geometry.
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