In Gelman and Hill 2007 (http://www.stat.columbia.edu/~gelman/arm/), they mention that adding an $\epsilon \sim N(0,\sigma^2)$ term to a Poisson regression can be used to account for overdispersion.
On a number of different sites I've seen this mentioned, but all refer you back to Gelman and Hill for an explanation of this, and the text doesn't address conceptually why this makes sense (although I believe them).
I understand that a two-parameter distribution (eg negative-binomial) may be used in place of a Poisson, and this makes sense to me: it has an extra parameter to capture variance. However, it is not at all obvious to me why simply adding an $\epsilon$ term should capture overdispersion, as the resulting Poisson will still have mean equal to its variance (as always). Can anyone clarify what's going on here and why this should account for overdispersion, as an alternative to using a negative binomial? And is there any simple way of interpreting how much overdispersion this $\epsilon$ should capture?