proper argument: fix a vertex $v_0$. in the Leibniz formula for the determinant, you sum over bijections $\sigma\colon\{\text{vertices except $v_0$}\}\rightarrow\{\text{edges}\}$ and i claim there is only such bijection where $v$ is incident to $\sigma(v)$ for each vertex $v\neq v_0$. any bijection where that doesn't hold only contributes a $0$ summand.
now, the point is that such a bijection is determined inductively by depth counted from $v_0$. any vertex of depth $1$ is adjacent to $v_0$ via a unique edge and has to be mapped to that edge as otherwise no vertex could be mapped to that ed…