I can't figure it out myself. But it seems that the claim must be wrong. The eigenvalues are not integral. For example, with $n=4$ the matrix is $$ A=\begin{pmatrix} 0 & 4 & 0 & 0\cr 1 & 0 & 3 & 0\cr 0 & 2 & 0 & 2 \cr 0 & 0 & 3 & 0 \end{pmatrix}. $$
The characteristic polynomial of this matrix is $\chi (t)=t^4-16 t^2 +24$, which has no integral roots. Am I overlooking something ?