Good question! That's a ggplot2-specific convention for referring to values that are calculated by the geom or stat. You can see a list of the ones available by looking under "Computed Variables" in the documentation for a geom or stat (for this case, where stat_count() is doing the work: https://ggplot2.tidyverse.org/reference/geom_bar.html#computed-variables).
The twist is that since the recent release of ggplot2 3.0, the awkward ..s can be replaced by the new stat() function. So a more modern version of your code would be:
ggplot(iris) +
aes(x = species) +
geom_bar() +
geom_text(stat = 'count', aes(label = stat(count), vjust = -0.2))
For more info, see: https://ggplot2.tidyverse.org/reference/stat.html
(If by any chance the talk of geoms and stats above left you confused, your best bet is to work through the Data Visualization chapter of R for Data Science: http://r4ds.had.co.nz/data-visualisation.html)