I'm having trouble understanding the end of the proof of this theorem. I will provide screenshots. It seems like they use Kronecker's lemma (Theorem 2.5.9) at the end but I do not quite understand it. Furthermore, I do not understand why there is a $p^{-1}$ factor in the estimate for $|\mu_m|.$
The factor of $p^{-1}$ in the estimate for $|\mu_m|$ is spurious. In the last argument, $X_i$ and $X$ should either both be $X_m$ or both be $X_1$. At the end, the Stolz-Cesaro theorem [1] is implicitly used. It is just as easy to repeat its proof:
Write $$z_m:=E(|X_m|^p \, ; \, |X_m|>m^{1/p}) =E(|X_m|^p \, {\bf 1}_{\{|X_m|>m^{1/p}\}} ) \,.$$ Given $\epsilon>0$, dominated convergence implies that there exists $m_\epsilon$, such that $z_m<\epsilon$ for all $m \ge m_\epsilon$. Then $$\limsup_{n \to \infty} \: n^{-1/p} \sum_{m=1}^n |\mu_m| \le \limsup_{n \to \infty} \: n^{-1/p} \sum_{m=m_\epsilon}^n z_m m^{-1-{1/p}} $$ $$ \le \epsilon n^{-1/p} \sum_{m=m_\epsilon}^n m^{-1-{1/p}} \le C_p \epsilon\,.$$ Since $\epsilon>0$ is arbitrary, the $\limsup$ on the left equals zero.
[1] https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem