It’s easier to answer this question if you leave out the six states that didn’t elect any Republicans in 2000; after all, they didn’t have any to throw out. If you also remove New Hampshire and South Dakota, where the percentage of Republicans elected dropped to 0 from 100 — New Hampshire only has two seats in the House and South Dakota has one — a pattern starts to appear.
At first sight, this appears as a case of removing outliers, which many statisticians recommend. Except that the data omitted were not outliers. Indeed, when both x- and y-variables are bounded (between 0% and 100% share of the House seats; between -100% and +100% change in share), there can be no extreme values.
In effect, when the author eliminated those eight points, he followed the "emergent pattern" theory, by which I mean the notion of removing data until a pattern "emerges". (By the way, emergence is now a science, as expounded here.) If enough data is removed, one can produce any pattern as one pleases. One can find subsets of data to support a hypothesis of positive linear, flat linear or quadratic, as shown below.
Focusing now on the full data set on the upper left corner, one is hard pressed to conclude that a positive correlation exists between the two variables. In particular, most states experienced no changes in the share of House seats, and in these states, the income growth ranged from under 20% to over 40%, which is pretty much the extent of variability across the full data set.