The relationship between variance and rms value follows directly from the definition of variance. For a continuous random variable, it is defined as:
$$
\sigma^2 = E\left(\left(X - \mu\right)^2\right)
$$
Where $\mu$ is the mean of the random variable $X$. Noise processes are typically modeled as zero-mean, so this simplifies to:
$$
\sigma^2 = E\left(X^2\right)
$$
So the variance $\sigma^2$ is defined as the expected value, or the mean, of $X^2$.
How does this relate to $X$'s RMS value? RMS stands for "root mean squared", or "the square root of the mean of the square" of the random variable. By inspection, you can see the relationship you asked about:
$$
\text{RMS} = \sqrt{E(X^2)} = \sqrt{\sigma^2} = \sigma
$$
$\sigma$ is also known as the standard deviation of the random variable.