Characteristic polynomial of a tree

Question

I'm trying to understand the article: László Lovász, Jozsef Pelikan, On the Eigenvalues of Trees, Periodica Mathematica Hungarica, March 1973. I'm not sure I fully understand the proof of Lemma 1: if $e$ is an edge of a forest $G$, $$f_G(\lambda)=f_{G-e}(\lambda)-f_{G-[e]}(\lambda)$$ where $f_G$ denotes the characteristic polynomial of $G$, $G-e$ is $G$ with edge $e$ removed, and $G-[e]$ is $G$ with $e$ removed and its endpoints. I've tried to write the adjacency matrix and to expand the determinant but I still don't understand it.