I am reading a textbook which casually mentions that the matrices
$$A = \begin{pmatrix} a & b \\ c & a + b - c \end{pmatrix}$$ and
$$B = \begin{pmatrix} a+b & - b \\ 0 & a-c \end{pmatrix}$$ are similar because
$$ B = \begin{pmatrix} 1 & 0 \\ 1 & -1 \end{pmatrix} A \begin{pmatrix} 1 & 0 \\ 1 & -1 \end{pmatrix}^{-1},$$ which allows one to conclude that the eigenvalues of $A$ are $a+b,a-c$.
Its easy to verify by direct calculation that this is true, but it feels pulled out of a hat and I prefer to have proofs which "make sense" to me, i.e., proofs where I understand where each step "came from." I'm wondering if anyone can explain this similarity relation to me on that level.