Let 
$$T(z)=\sum_{1}^\infty \frac{n^{n-1}x^n}{n!}.$$
This is known as the exponential generating function of unordered rooted trees, It follows from the <a href="https://www.math.purdue.edu/~eremenko/Pdf/burmann.pdf">Burmann-Lagrange formula</a> (example 1 on p. 3) that this function solves
$$T(x)=xe^{T(x)}.$$
Your sum is $y=T(1/e)$. So we have to solve the equation
$$y=e^{y-1}.$$
This has a root $y=1$ which is multiple (of multiplicity $2$).
This means that the graph of the LHS is tangent to the graph of the RHS
at the point $(1,1)$. Since the LHS is linear and RHS is convex,
our equation has unique solution, namely $y=1$, which proves your formula.