$||f + g||_{L^p} = ||f ||_{L^p} +||g||_{L^p} $ for $p\in\mathbb{R}^+\setminus\{ 1\}$ and $f,g\geq 0$ $||f + g||_{L^p} = ||f ||_{L^p} +||g||_{L^p} $ for $p\in\mathbb{R}^+\setminus\{ 1\}$ and $f,g\geq 0$, then $f = Cg$ for some non-negative constant $C$.
First assume $||f ||_{L^p} +||g||_{L^p} = 1$, then $(\cdot)^p$ on the interval $(0,\infty)$ is strictly convex for $p>1$, thus we have
$$(f+g)^p = (\frac{f}{||f||} ||f|| + \frac{g}{||g||} ||g||)^p < \frac{f^p}{||f||^p} ||f|| + \frac{g^p}{||g||^p} ||g|| ,$$
integrate both side
$$1 = \int_X (f+g)^p dx <  \int_X \frac{f^p}{||f||^p} ||f|| + \frac{g^p}{||g||^p} ||g|| dx = ||f|| + ||g|| = 1.$$
Clearly the above inequality can not be true, thus we have to have
$$\frac{f}{||f||} = \frac{g}{||g||},$$
which is the only way to get equality from a strictly convex function. It implies
$$f = Cg$$ 
for some non-negative constant $C$.
The same convex argument can be applied to $0<p<1$ since $(\cdot)^p$ would be strictly concave. And to get equality from a strictly concave function, we must have
$$\frac{f}{||f||} = \frac{g}{||g||}.$$
For $f$ and $g$ in general, replace with$\frac{f}{||f||+||g||}$ and $\frac{g}{||f||+||g||}$, we have
$$\left|\left|\frac{f}{||f||+||g||} + \frac{g}{||f||+||g||}\right|\right| = 1.$$
 A: There is no mistake. The strict convexity inequality $$\phi(tx+(1-t)y)< t\phi(x)+(1-t)\phi(y) ,\quad 0<t<1$$
turns into equality if and only if  $x=y$. And this precisely corresponds to one function being a multiple of the other. 

(old answer)
The case $p<1$ goes like this. The set where both $f$ and $g$ are zero can be ignored.  On the rest, $f+g$ is strictly positive, so we can manipulate with its negative powers: 
$$
\int (f+g)^p  =  \int f(f+g)^{p-1} +  \int g(f+g)^{p-1} \ge \|f\|_p \| (f+g)^{p-1}\|_q +  \|g\|_p \| (f+g)^{p-1}\|_q 
$$ 
where the second step is the reverse Hölder's inequality, and
 $q$ is the conjugate exponent to $p$ ($q$ is negative!). This simplifies to $\|f+g\|_p^p \ge (\|f\|_p+\|g\|_p) \|f+g\|_p^{p-1} $ as wanted. So, if equality holds, it also holds in the two instances of the reverse Hölder's inequality  above. Hence
$|g|^p$ and $|f|^p$ are both constant multiples of $(|f+g|^{p-1})^q$, which makes them collinear vectors in $L^1$. Hence, $f$ and $g$ are collinear. 
