However, the figure represents spending and growth in the same year. Suppose you take a person whose income goes up and down from year to year and look at spending in some area, like food and drink. You'd expect a substantial positive correlation between income and spending--when people have more to spend, they spend more. The same would apply to governments--when tax revenue rises, governments are likely to introduce new programs and give government employees generous raises; when they fall, they'll cut programs and impose hiring and pay freezes.
So if you want to know whether government spending affects growth, you need to look at spending in year x and growth in year x+1. The Keynesian argument is that there should be a positive relationship (government spending saves or creates jobs, people with jobs spend money, and the economy grows), the "austerian" argument is that there should be a negative relationship (government spending inhibits private spending and investment). The figure:
The relationship is weaker, but still pretty evident (and statistically significant). But growth may also depend on last year's growth, not in a causal sense, but in the sense that much of what was happening last year, for good or ill, will probably happen this year too. If you add last year's growth as a control:
The estimate for last year's growth (lgrow) is statistically significant (t=6), and the estimate for last year's spending growth (lgov) is nowhere near significant (t=0.5). That isn't proof (or even evidence) that government spending doesn't matter--it basically means that anything is possible: it could help, hurt, or make no difference.
Krugman says, "you can, if you like, try to argue that this relationship is spurious, maybe not causal." Actually, I liked his original figure, since I agree with Krugman on economic policy. But thinking about the possibility of spurious correlation isn't a matter of liking--it should be pretty much automatic.