Let us show it in two steps:
$$
f_n\to f\;\;\text{in } L^p\;\;\Longrightarrow\;\;|f_n|^p\to |f|^p \;\;\text{in }L^1\;\;\Longrightarrow \;\;\int |f_n|^p\,\mathrm d\mu\to\int |f|^p\,\mathrm d\mu.
$$
The last implication should be pretty straightforward. For the first implication we might use that for $x,y\in\mathbb{R}$ one has
$$
||x|^p-|y|^p|\leq p(|x|^{p-1}+|y|^{p-1})|x-y|
$$
and hence
$$
\begin{align}
\int ||f_n|^p-|f|^p|\,\mathrm d\mu & \leq \int p(|f_n|^{p-1}+|f|^{p-1})|f_n-f|\,\mathrm d\mu \\
&=\int p|f_n|^{p-1}|f_n-f|\,\mathrm d\mu+\int p|f|^{p-1}|f_n-f|\,\mathrm d\mu\\
\end{align}
$$
which by the use of Hölder's inequality is less than or equal to
$$
p\left(\int |f_n|^{q(p-1)}\,\mathrm d\mu\right)^{1/q}\left(\int |f_n-f|^p\right)^{1/p}+p\left(\int |f|^{q(p-1)}\,\mathrm d\mu\right)^{1/q}\left(\int |f_n-f|\,\mathrm d\mu\right)^{1/p}\\
$$
where $q>1$ is chosen such that $\frac1p +\frac1q=1$. This simplifies to (recall that $q(p-1)=p$)
$$
p\left[\left(\int |f_n|^p\,\mathrm d\mu\right)^{1/q}+\left(\int |f|^p\,\mathrm d\mu\right)^{1/q}\right]\left(\int |f_n-f|^p\,\mathrm d\mu\right)^{1/p}.
$$
The last factor tends to $0$ as $n\to\infty$ by assumption, and thus we only need to argue that
$$
\sup_n\int |f_n|^p\,\mathrm d\mu<\infty.
$$
This i'll leave for you to prove.
