Why does Stouffer's method work? It seems like a fairly straightforward question, but when I really think about it, Stouffer's method doesn't make sense to me. This is why: 
Assume a two-tailed hypothesis. You first calculate $z_i$ from $p$-values. So let's take a fairly simple example. Let's take two $p$-values of $0.05$. This means that $z_1$ and $z_2$ are both $\approx1.96$. According to Stouffer's method, $z_1$ and $z_2$ are combined such that:  
$$
Z = \frac{\sum\limits_{i=1}^kZ_i}{\sqrt{k}} = \frac{1.96 + 1.96}{\sqrt{2}} = 2.77
$$
This $z$-score then gets converted to a $p$-value once again, resulting in a $p$-value of $0.005$, whereas the $p$-values from each $z_i$ individually is about $0.05$. 
In this sense, it seems as though Stouffer's test artificially changes the resultant $p$-value to a value dissimilar to the $p$-values of each $z_i$, which to me, doesn't make sense.
Am I misunderstanding this test or can someone help me understand how / why it works? 
 A: The higher overall sample size leads to a higher power and thus to a smaller p value (at least if the working hypothesis is supported by the data). 
This is usually the main point of any meta analysis: multiple weak evidences supporting a hypothesis are combined to strong evidence for it.
A: For simplicity think in terms of a test on means.  Suppose under H0 the treatment effect is zero, so that each z value is a weighted estimate of the treatment effect θi.  Stouffer's method gives an unweighted average of these treatment effects so will give a more precise estimate (and hence smaller p-value) than each separate z value.  This unweighted estimate of the treatment effect is biased but a weighted Stouffer's method is possible, and if the weights are proportional to 1/standard error(θi) the treatment effect estimate is unbiased.  This only makes sense however if the separate z values are measures of the same quantity.  An advantage of Stouffer's and Fisher's methods is that they can also be applied to meta-analyses where different response variables have been chosen - so they can't be averaged - but where a consistency of direction of effect can still be discerned.
A: Think of it from a meta-analysis point of view:  If there was no effect ($H_0$), $p$ values would be equally distributed between 0 and 1.  So if you get $p<0.1$ in more than 10% of all single analyses (potentially many of them), this can amount to the conclusion that $H_0$ probably should be rejected.
I do not even see a problem for two-tailed tests:  In this case the result should be interpreted as: It is unlikely that the true mean is 0 (in the example of a gaussian around 0), but I cannot tell (from either the previous or the combined $p$ value) if the true mean is above or below it.
A: I think it'd be fine to combine 2-tailed results because that means that the result would amount to zero (if there is evidence that the treatment enhances [right-tail] the disease of a patient but also evidence that it worsens [left-tail], the net result is no evidence towards a particular hypothesis since they cancel out and more observations are needed.
