The exchange interaction is a type of interaction between identical particles caused by permutation symmetry. In other words, it can be loosely thought of as a quantum statistical effect which results in a contribution to the Hamiltonian of a many-particle system (not directly from any fundamental forces). one of the most common exchange interaction is the Heisenberg interaction, where the Hamiltonian of 2 interacting spinful paricles reads:
$$
H_{12} = J\vec{S_1} \cdot \vec{S_2} = J(S_1^x \otimes S_2^x+S_1^y \otimes S_2^y+S_1^z \otimes S_2^z)
$$
where $J$ is a constant that set up the exchange energy scale, and $S_1,S_2$ are defined by Pauli matrices in their own Hilbert space.
A ferromagnetic chain (J<0) can be a result from the above exchange interaction. In the simplest case where the "chain" has only two spin-1/2 particles, $H_{12}$ reaches its minimum at triplet states, that is, fully magnetized in some direction. If $J>0$, its minimum energy occurs at singlet state where two spins anti-align to give zero magnetization. It's easy to check by diagonalizing $H_{12}$ and look at its eigen energy and corresponding eigen state. Longer macroscopic chain gives the same picture of magnetization.
Then you may wonder where does this $H_{12} \propto \vec{S_1} \cdot \vec{S_2}$ comes from at the first place. Well, there are many different physical systems that gives such exchange interaction. One canonical example is the Fermi Hubbard model, whereby Heisenberg type exchange interaction emerges due to the hopping of electrons between different sites and the on-site Coulumb repulsion between two anti-aligned electrons. This can be proved by 2nd order perturbation theory of the Hamiltonian:
$$
H = H_0 + H_1
$$
where $H_1$ describe the hopping between sites (kinetic energy) and $H_0$ the on-site electrostatic energy:
$$
H_0 \propto I \sum_{i,\sigma}n_{i,\sigma}n_{i,-\sigma}
,\;\;\;
H_1 \propto \sum_{ij,\sigma} T_{ij} c_{i,\sigma}^\dagger c_{i,\sigma}
$$
If $H_1$ is small compared to $H_0$, and notice the equivalence between spin operator and creation/annihilation operator:
$$
S_i^z = \frac{1}{2} \sum_{\sigma} n_{i,\sigma},\;\;\; S_{i,\sigma} = c_{i,\sigma}^\dagger c_{i,-\sigma}
$$
the 2nd order perturbation theory will give you an effective Hamiltonian that has exchange interaction $H \propto \sum_{ij} S_i\cdot S_j + const$. For details you may refer to this pedagogical paper: https://aapt.scitation.org/doi/10.1119/1.10537
