The conceptual reason is that different sources have uncorrelated phases, so that at each frequency, they will randomly either add or subtract from the sound pressure. Adding these sources together is like taking steps in a random walk.
Imagine that for each step, you randomly choose to move to the left or the right by a distance of 1. After $N$ steps, the expected change in your position is 0. Correspondingly, for sound, the average perturbation in the air pressure is 0.
However, for the random walk, after $N$ steps, the expectation of the square of the change in position is $N$. That is, the squares of the step lengths are additive, and correspondingly for sound, the squared amplitudes are additive.
Mathematically, we can write this as follows. Let $P_i$ be the sound pressure perturbation contributed by each source $i$. The sound intensity contributed by that source is $I_i=\langle P_i^2\rangle$, where the angle brackets denote the statistical expectation. However $\langle P_i\rangle = 0$, since sound equally contributes positive and negative pressure perturbations. Also, since different sources are uncorrelated, $\langle P_i P_j\rangle=0$ for $i\neq j$.
Adding together the contributions of all of these sources, the expectation of the total sound pressure perturbation is
$$\left\langle \sum_i P_i\right \rangle = \sum_i\langle P_i\rangle = 0.$$
However, the expectation of the squared sound pressure perturbation is
$$\left\langle \left(\sum_i P_i\right)^2\right \rangle = \sum_{i,j} \langle P_iP_j\rangle = \sum_{i=j}\langle P_iP_j\rangle + \sum_{i\neq j}\langle P_iP_j\rangle.$$
The second term vanishes since different sources are uncorrelated, while the first term is just the sum of the sound intensities. That is,
$$\left\langle \left(\sum_i P_i\right)^2\right \rangle = \sum_{i}\langle P_i^2\rangle = \sum_i I_i.$$
