- Some explain it as something that works like the Poisson distribution but has an additional parameter, allowing more freedom to model the true distribution, with a variance not necessarily equal to the mean
- Some explain it as a weighted mixture of Poisson distributions (with a gamma mixing distribution on the Poisson parameter)
Mathematically one obtains Negative binomial by integrating the Poisson distribution over Gamma-distributed weights, see Gamma-Poisson mixture. This mathematical fact remains regardless of whether we accept it as the justification for using the distribution or not.
Poisson distribution is a rather natural choice when talking about counting reads arising from DNA sequencing (one could use binomial, but given that one sequences only a small fraction of reads/DNA obtained from the sample, the difference is negligible, and we can use whatever seems more convenient.) We are also sure that the parameter of this Poisson distribution varies, although the reason for this variation depends on the exact nature of the experiment - e.g., it can be variation due to
- replicating the same experiment several times
- the reads originating from different cells with somewhat different properties
- comparing the numbers of reads corresponding different genes
- genes having different chemical structure and therefore amplified differently by PCR or some reads more likely to make their way to the sequencing machine
- the student/postdoc preparing the libraries not being very careful/consistent
In other words, we are sure that the variation exists (and we do observe it experimentally), but we don't know exactly where it comes from, and we cannot directly know what probability distribution describes it. We couldn't model it using the normal distribution, since the Poisson parameter should be positive, so we use the Gamma distribution, because it is "almost like normal", but with non-negative support... but we could have also used log-normal or something else. As long as we are not looking for the fine biological effects that could turn out to be artifacts of the particular distribution we use, anything that is computationally convenient is good.
Note that, besides the flexibility provided by an extra parameter, negative binomial has a thicker tail than the Poisson distribution, making it less sensitive to outliers. This provides an additional motivation for using this distribution: it allows more robust inference.