Why are different rms noise sources added in this way?
If the noise sources are uncorrelated, then the power (variance) of its sum equals the sum of their powers (variances), because all the co-variances between them are zero, and only the variances remain.

An example with two noise sources (with zero mean):

\$\sigma_N^2 = E[N^2] = E[(X+Y)^2] = E[X^2]+E[Y^2]+2E[XY] = \sigma_X^2 + \sigma_Y^2 +2\sigma_{XY}\$

If \$X\$ and \$Y\$ are uncorrelated then \$\sigma_{XY}=0\$ and \$\sigma_N^2 = \sigma_X^2 + \sigma_Y^2 \$, thus:

\$\sigma_N = \sqrt{\sigma_X^2 + \sigma_Y^2} \$, where \$\sigma_N\$, \$\sigma_X\$ and \$\sigma_Y\$ are the RMS values of \$N\$, \$X\$ and \$Y\$.

It can be demonstrated for any number of sources, if required. The result ends up being the same: power of the sum equals the sum of the powers.