In Thorpe's "Introduction to Optimal Transport" on page 18 in Theorem 2.7. we have that:
$$ \sum_{i=1}^nc(x_{i},T(x_{i}))=\sum_{ij}c_{ij} \pi_{ij} \geq \sum_{ij}c_{ij} \pi^\dagger_{ij}=\sum_{i=1}^nc(x_{i},T^{\dagger}(x_{i})) $$
I don't understand why that just works if $\pi^\dagger_{ij}$ is a permutation matrix?
And I also can't find any other proofs online for showing that the Monge and Kantorovich problems are equivalent in the discrete case.
I assume you mean the equalities and not the inequality?
If you mean the inequality, this holds just because $\pi$ can be any particular coupling while $\pi^\dagger$ is the optimal coupling.
The last equality follows largely from Theorem 2.6 which states the extremal points, such as the minimal coupling, will be a permutation matrix. Then a permutation matrix is a type of mapping, hence $\pi^\dagger$ is equal to a mapping, which will be optimal, denoted by $T^\dagger$.
There are some more subtleties such as the convexity of the set and the beginning of the proof does some extra work to show the existence of a solution to the Kantorovich formulation, but I think this is all that's happening at a high level.