Binary Integer Programming question - what graph problem is represented
This answer assumes that $c_{ij} \geq 0$.

The first two sets of equations guarantees that $x_{ij}$ is a permutation matrix. It defines a permutation $\pi$ on $\{0,\ldots,n-1\}$ in the following way: $\pi(j) = i$ if $x_{ij} = 1$.

The set of inequalities is a logical implication: if $x_{ik}=x_{j(k+1)}=1$ then $z_{ijk} = 1$. That is, if $\pi(k) = i$ and $\pi(k+1) = j$ then $z_{ijk}=1$.

Since $c_{ij} \geq 0$ and the objective is to minimize $\sum_{ijk} z_{ijk} c_{ij}$, we want to have $z_{ijk} = 0$ unless we are forced to take $z_{ijk} = 1$. Therefore $z={ijk} = x_{ik} \land x_{j(k+1)}$. This means that the objective function is
$$
\min_\pi \sum_{k=0}^{n-1} c_{\pi(k)\pi(k+1)}.
$$
This is the problem known as minimum directed Hamiltonian circuit.