The triangle inequality (Minkowski) is proven using Hölder. To show what you want, you can repeat that argument:

(note that $q(p-1)=p$)

$$
\|f\|_p+\|g\|_p=\|f+g\|_p=\left(\int|f+g|^p\right)^{1/p}=\left(\int|f+g|\,|f+g|^{p-1}\right)^{1/p}\leq\left(\int|f|\,|f+g|^{p-1}+\int|g|\,|f+g|^{p-1}\right)^{1/p}
\leq\left(\|f\|_p \left(\int|f+g|^p\right)^{1/q}+\|g\|_p \left(\int|f+g|^p\right)^{1/q}\right)^{1/p}
=\left((\|f\|_p+\|g\|_p)\left(\int|f+g|^p\right)^{1/q} \right)^{1/p}
=\left(\|f+g\|_p \|f+g\|_p^{p/q} \right)^{1/p}
=\left(\|f+g\|_p \|f+g\|_p^{p-1} \right)^{1/p}
=\left( \|f+g\|_p^{p} \right)^{1/p}
=\|f+g\|_p=\|f\|_p+\|g\|_p
$$

In particular we have equality in the two Hölder inequalities we used in the middle (the second "$\leq$"). Equality in Hölder occurs only when the $p$ and $q$ powers of the two factors are linearly dependent. This means that there exist constants $\alpha,\beta$ such that
$$
|f|^p=\alpha|f+g|^{q(p-1)},\ \ |g|^p=\beta|f+g|^{q(p-1)}
$$
almost everywhere.

So $|g|^p=\gamma|f|^p$ a.e. for some constant $\gamma$, and then $|g|=\delta|f|$ a.e. for yet another constant. Then
$$
\frac{|f|}{\|f\|_p}=\frac{|g|}{\|g\|_p}\ \ \ \text{a.e.}
$$