Generic stuff on Stallings' foldings and immersions
A map of graphs $f:A\rightarrow B$ is an immersion if it is locally injective (injective at the vertices). An example of an immersion is if you take a square and pinch two opposite sides together to get a line with a loop at each end. Then this is not an injection, but it is an immersion.
The proof of this lemma is based on an really neat idea, due to John Stallings. The idea basically says that instead of looking at covers to understand subgroups of $\pi_1(A)$, $A$ a graph, we can look at immersions (that is, subgraphs). As $\pi_1(A)$ is free, this gives an elegant, and extremely important, way of looking at subgroups of free groups. The key phrase is ""Stallings' foldings". These are the "folds" mentioned in the paper (see also section 2.4 on p528/p12), and were introduced in the (extremely readable) paper "J. Stallings, Topology of finite graphs, Invent. Math. 71 (1983), 551-565" (reference [Sta] in your paper).
Stallings' proves that every map of graphs $f:A\rightarrow B$ factors as $A\rightarrow C\rightarrow B$ where $C\rightarrow B$ is an immersion and $A\rightarrow C$ is a "folding" map. He also proves that this folding map is unique, although the individual folding moves may not be.
Stuff specific to the paper
In the proof in this paper we are considering a map $f:A\rightarrow A$. This factors as folds $p:A\rightarrow C$ and an immersion $f':C\rightarrow A$. Therefore, if the sentence "The only fold that can take place is between the initial and terminal ends of $e_n$" is true then the result follows from standard results about Stallings' foldings. So, (a) do you understand this sentence, and (b) do you understand that this is the key sentence and everything else is standard (for some value of "standard"...)?