A PCP is a probabilistically checkable proof, and the PCP verifier is the algorithm that checks it. In our case, we are given an instance of Unique Label Cover which is promised to be either at least $(1-\eta)$-satisfiable (Yes instance) or at most $\gamma$-satisfiable (No instance). The (polynomial size) proof, which is similar to a witness in an NP verifier, purports to show that the instance is a Yes instance. It is given by a sequence of bits. The verifier is a probabilistic algorithm which checks the proof. If the instance is a Yes instance then there is a proof which convinces the verifier with probability at least $c$ (the completeness), and if the instance is a No instance then every proof convinces the verifier with probability at most $s$ (the soundness).
This particular verifier chooses two bits $\pi_i, \pi_j$ of the proof $\pi$ according to some complicated distribution, and checks that $\pi_i \neq \pi_j$. Every Yes instance has a proof such that $\Pi[\pi_i \neq \pi_j] \geq c$, and for No instances, all proofs satisfy $\Pr[\pi_i \neq \pi_j] \leq s$.
You can encode this verifier as an instance of Max Cut. The vertices $v_1,\ldots,v_N$ correspond to the bits of the proof $\pi_1,\ldots,\pi_N$, and two vertices $v_i,v_j$ are connected with an edge whose weight is the probability that the verifier chooses to check the two bits $\pi_i,\pi_j$. A cut in the graph can be though of encoding a proof: vertices on one side of the cut are 0 bits, and vertices on the other size are 1 bits. Under this encoding, an edge $(v_i,v_j)$ is cut iff in the corresponding proof, $\pi_i \neq \pi_j$.
The upshot is that if the Unique Label Cover instance is a Yes instance, then the Max Cut instance has a cut whose value is at least $c$, whereas if the Unique Label Cover instance is a No instance, all cuts of the Max Cut instance have value at most $s$. Therefore any approximation algorithm for Max Cut whose approximation ratio is better than $c/s$ would be able to solve Unique Label Cover, which is UGC-hard.