Why does ε-differential privacy protect the subset of 1/ε edges in terms of graphs?
I think this is a reference to group privacy. See Theorem 2.2 in the Dwork-Roth book.

If you have $(\varepsilon,0)$-differential privacy for changing 1 edge, then you have $(1,0)$-differential privacy for changing $1/\varepsilon$ edges.