What they show is that even when you control for overall state ideology (Democratic states have higher taxes, and really Democratic states have much higher taxes), union density increases tax rates. Increasing union density from 10% of the workforce, as in Nebraska, to 25%, as in Hawaii and New York, increases the tax burden by about one and a quarter percentage points of state personal income. For a sense of scale, the mean tax burden was 10.0% of personal income in 2008, and the standard deviation was 1.23, so this is essentially a standard deviation increase.
By passing right-to-work, Indiana could expect its unionization rate to drop anywhere between four and nine percentage points, taking in the range of values observed in other right-to-work states. This change would decrease Indiana’s tax burden in the long run by between a third and three-quarters of a percentage point of personal income. The predicted effects in New Hampshire would be slightly less, since New Hampshire is slightly less unionized than Indiana.
It’s difficult to make causal inferences based on these kind of statistical correlations, but the underlying theory here seems pretty clear. Under both the conservative “greedy public servants demand high pay for themselves” theory and the progressive “unions are a crucial counterweight to the political influence of the rich” theory, higher levels of public spending are a major political consequence of unionization. One alternative interpretation that I would like to see statistically tested would be that richer places have higher burdens (which is the general global and historical trend) and right to work laws happen to have proliferated in poor southern states 50 years ago.