The most complete explanation of what the dispersion means from a scientific point of view is probably in the edgeR glm paper:
See the first section of Results in conjunction with the first section of Methods. That article characterized sqrt(dispersion) as the "biological coefficient of variation (BCV)", and that is the terminology we have used since in the edgeR articles and documentation. The BCV is the relative variability of expression between biological replicates.
If you estimate dispersion = 0.19, then sqrt(dispersion) = BCV = 0.44. This means that the expression values vary up and down by 44% between replicates.
An important point, that is easy to miss, is that the BCV measures the relative variability of true expression levels, not the variability of measured expression levels. The BCV represents the relative variability that you would observe if you were able to measure the true expression levels perfectly in each RNA sample, even though one can't actually do that. It represents the variability that remains after the Poisson variability from sequencing has been removed.
To repeat, BCV does not represent the variability between observed expression levels. It is the variability of true expression levels. You cannot measure BCV using an undergraduate formula from the observed counts or RPKM values.
Afternote: I have just noticed that you asked the same question at the same time on Biostars:
I also see that you have previously posted a number of questions about edgeR on Biostars but not on Bioconductor, and some of those questions went unanswered. Please be aware the that edgeR authors try hard to answer questions on Bioc, but we don't have the time or resources to monitor every possible forum.
That looks like the DESeq2 paper. We define the dispersion parameter in the first sentence of the section in the main text where we introduce it:
"Within-group variability, i.e., the variability between replicates, is modeled by the dispersion parameter alpha, which describes the variance of counts via..."
The dispersion parameter links the variance and mean of the count for the negative binomial distribution.