As @DavidLHarden explains in the link that you gave, this theorem is proved by attending to the $p$-part and $p'$-part separately.

For the $p'$-part the result follows from the following theorem of Burnside:

Let $\psi$ be a $p'$-automorphism of the $p$-group $P$ which induces the identity on $P/\Phi(P)$. Then $\psi$ is the identity automorphism of $P$.

This is the result that Geoff refers to in his comment above. It is discussed and proved in Section 5 of Gorenstein's Finite Groups, specifically Theorem 1.4 of that section.

I do not know of a reference for the $p$-part of the proof. You should certainly look at the paper by Neumann that Geoff mentions, however if I understand that proof correctly it only proves your bound for $|Out P|$, rather than $|Aut P|$. On the other hand Neumann is considering a much more general setting than just $p$-groups.

As @DavidLHarden explains in the link that you gave, this theorem is proved by attending to the $p$-part and $p'$-part separately.

For the $p'$-part the result follows from the following theorem of Burnside:

Let $\psi$ be a $p'$-automorphism of the $p$-group $P$ which induces the identity on $P/\Phi(P)$. Then $\psi$ is the identity automorphism of $P$.

This is the result that Geoff refers to in his comment above. It is discussed and proved in Section 5 of Gorenstein's Finite Groups, specifically Theorem 1.4 of that section.

I do not know of a reference for the $p$-part of the proof. You should certainly look at the paper by Neumann that Geoff mentions, however if I understand that proof correctly it only proves your bound for $|Out P|$, rather than $|Aut P|$. On the other hand Neumann is considering a much more general setting than just $p$-groups.