A consequence of edge-criticality Let $G$ be $\Delta$-edge-critical (that is, $G$ is $\Delta + 1$-edge-chromatic and removing any edge of $G$ gives a subgraph which is at most $\Delta$-edge-chromatic, where $\Delta$ is the max degree of $G$). Let $d(x) + d(y) = \Delta + 2$, and let $\pi$ be a $\Delta$-coloring of $G-xy$. Let $S$ be the set of colors used in $\pi$ The article I'm reading goes on to claim without proof that $C_{\pi}^{\prime}(x)\cap C_{\pi}^{\prime}(y) = \emptyset$, and furthermore that $C_{\pi}^{\prime}(x)\cup C_{\pi}^{\prime}(y) = S$. (That is, there is no color absent at both $x, y$, but every color is absent at exactly one of $x,y$.) The first statement _seems_ to be proven by saying that if there were a color absent at both $x$ and $y$, removing the edge $xy$ from $G$ would not change the edge-chromaticity, but I don't trust my intuition. On the second statement I'm totally lost, because I can't even begin to see how applying the sum of the degrees helps.

Your intuition on the first statement is correct, but why 'intuition'? If a color is absent at both $x$ and $y$ you can use it to color $xy$ and get a $\Delta$-coloring of $G$. This is logic.

Now note that in $G-xy$ the degreesum of $x$ and $y$ is exactly $\Delta$, i.e. the number of colors. If no color is absent, every color is used exactly once.