How does averaging increase the Signal to Noise ratio?
Intuitively this is true, because averaging a zero mean noise processes approximates its expectation value - which is zero. More rigorously:

If the signal $x$ that you want to observe (estimate, actually) is constant for all observations $y$ we can write the $n\text{th}$ observation as
$$
y_n = x + r_n
$$
where $r_n$ is the noise which is different for every observation, of course. The SNR $\gamma_1$ of $y_n$ is
$$
\gamma_1=\frac{\operatorname{E}[|x|^2]}{\operatorname{E}[|r_n|^2]} 
$$
where $\operatorname{E}$ is the expectation operator. By averaging $N$ observations of $y_n$ we get
$$
\hat x = \frac{1}{N}\sum_{n=1}^N( x+r_n) = x + \frac{1}{N}\sum_{n=1}^N r_n
$$
The SNR $\gamma_2$ of the averaged observations is
$$
\gamma_2=\frac{\operatorname{E}[|x|^2]}{\frac{1}{N^2}\operatorname{E}[|\sum_{n=1}^N r_n|^2]} 
$$
To simplify that expression, we assume that the noise sequences $r_n$ are uncorrelated (1). In that case, the expectation of the squared sum of $r_n$ is equal to the sum of the expectations of the squared $r_n$, i.e.
$$
\operatorname{E}\left[|\sum_{n=1}^N r_n|^2\right] = \sum_{n=1}^N \operatorname{E}\left[|r_n|^2\right]=N\operatorname{E}\left[|r_n|^2\right]
$$
Inserting this in the expression for $\gamma_2$ yields
$$
\gamma_2=N \frac{\operatorname{E}[|x|^2]}{\operatorname{E}[|r_n|^2]}
$$
Comparing the SNR $\gamma_1$ of the single observation and the SNR $\gamma_2$ of the averaged observation we see that averaging increases the SNR by a factor of $N$.

(1) This holds for a Gaussian noise process, for example. Right now I'm not sure for what classes of stochastic processes it holds. Comments on this are welcome.