A purely graph-theoretic explanation of the reduction from Unique Label Cover to Max-Cut
Let me see if I can clarify this, on a high level. Assume the UG instance is a bipartite graph $G = (V \cup W, E)$, bijections $\{\pi_e\}_{e \in E}$, where $\pi_e\colon \Sigma \to \Sigma$, and $|\Sigma| = m$. You want to construct a new graph $H$ so that if the UG instance is $1-\delta$ satisfiable, then $H$ has a large cut, and if the UG instance is not even $\delta$-satisfiable, then $H$ has only very small cuts. 

The graph $H$ contains, for each vertex in $W$, a cloud of $2^m$ points, each labeled by some $x \in \{-1, 1\}^\Sigma$. The intention is that you should be able to interpret a long code encoding of the labels of $W$ as a cut of $H$. Recall that to encode some $\sigma \in \Sigma$ with the long code, you use a boolean function $f\colon \{-1, 1\}^\Sigma \to \{-1, 1\}$; in particular it is the dictator function $f(x) = x_\sigma$. Let's produce a cut $S\cup T$ (i.e. bi-partition of the vertices) from the long code encoding as follows. If $w \in W$ has a label encoded by the boolean function $f$, go to the the cloud of vertices in $H$ corresponding to $w$, and put in $S$ all vertices in the cloud that are labeled by some $x$ for which $f(x) = 1$. All others go to $T$. You can do this backwards to assign boolean functions to all $w \in W$ based on a cut of $H$.

In order for the reduction to work, you need to be able to tell only by looking at the value of a cut $S\cup T$ whether the boolean functions corresponding to the cut are close to a long code encoding of some assignment of labels to $W$ that satisfies a lot of the UG constraints of $G$. So the question is what information do we get from the value of a cut $S \cup T$. Consider any two vertices $a$ with label $x$ in the cloud corresponding to $w$ and $b$ with label $y$ in the cloud corresponding to $w'$ (in the reduction we only look at $w$, $w'$ in different clouds). We said that the cut can be used to derive boolean functions $f_w$ and $f_{w'}$. Now if there is an edge $(a,b)$ in $H$, then $(a, b)$ is cut if and only if $f_w(x) \neq f_{w'}(y)$. Therefore, using only the value of a cut to tell if the boolean functions it induces are "good" is the same as having a test that, given boolean functions $\{f_w\}_{w \in W}$, only asks for what fraction of some specified list of pairs $((w, x), (w', y))$ we have $f_w(x) \neq f_{w'}(y)$. 

In other words, whenever Ryan says in the notes "test if $f_w(x) \neq f_{w'}(y)$", what he really means is "in $H$, add an edge between the vertex in the cloud of $w$ labeled by $x$ and the vertex in the cloud of $w'$ labeled by $y$". I.e. for every $v\in V$, every two of its  neighbors $w, w'$, and every $x,y \in \{-1, 1\}^n$, include the edge between the vertex in the cloud of $w$ labeled by $x\circ \pi_{v,w}$ and the vertex in the cloud of $w'$ labeled by $y \circ \pi_{v,w'}$, and assign the edge weight $(({1-\rho})/{2})^d(({1+\rho})/{2})^{n-d}$ where $d$ is the Hamming distance between $x$ and $y$. In this way the value of a cut divided by the total edge weight is exactly equal to the success probability of the test.