The article, unsurprisingly, is full of regression tables. I skimmed though it. Here are a couple of key paragraphs from the conclusion:
I showed a positive relationship between US troops and three social indicators across 148 countries during two two-decade periods, 1970–1990 and 1990–present. On the simplest level, countries with a heavy US troop presence had faster increases in life expectancy, faster reductions in rates of child mortality, and faster development of telephone lines per capita. Comparing the countries with a heavy US troop presence (250 or more troops per year) to those with essentially no US presence (five or fewer troops per year), the heavy presence countries experienced an additional decrease in children’s mortality rates of 21% points during 1970–1990 and 13% points 1990–present. Among low-income countries, gains in life expectancy during the first period averaged almost 10% (5.6 years) higher in heavy presence countries than in nil presence countries and 3.6% higher in the post-Cold War period. Increases in telephone lines per capita were four times larger in the heavy presence countries compared to nil presence countries during the Cold War and two times larger during the latter period.
Regression tests showed these relationships to be statistically significant, even when controlling for initial levels and growth rates of GDP per capita, conflict, economic aid, and other factors. A tenfold increase in US troops during a 20-year period in a typical host country is estimated in this paper to improve the reduction in children’s mortality by 2.2% points, improve life expectancy gains by 1% point, and increase the number of telephone lines by two per 100 people.
And here's the final paragraph:
The positive effect of US troop presence across over 148 countries is a new finding. However, much remains to be done. First, the mechanism of the developmental effect of hosted US forces is not known and also problematic since it was almost certainly unintentional at a tactical and strategic level. Although Mancur Olson [in his The Rise and Decline of Nations] theorized such a positive effect in his writings, the concept has not been carefully assessed with modern data and techniques until now. And yet, these results should be interpreted with some caution because the troops–growth relationship represents historical alliances, which is a far cry from normative justification for regime change [what? what does regime change have to do with anything here? --LFC]. In addition, the data are aggregated at the highest level possible. While it is beyond the scope of this paper to delve into causal explanations at the microlevel, growth theory suggests troops enhanced technology diffusion in some fashion, and indeed it may be that troop presence is simply an easily quantifiable proxy for overall US engagement. Nevertheless, whether directly or indirectly, the impact of US troops on global social development—during and since the Cold War—has been clearly positive, a fact that alone merits widespread recognition.
OK. Let's say this is right. Does this mean the U.S. should put more soldiers into more countries, in addition to the hundreds of thousands it already has? No, for two reasons. (1) This is objectionable on political grounds, in that, among other things, it may very well increase resentment and even hatred of the United States. American soldiers in Saudi Arabia might have indirectly given that kingdom some additional phone lines and life expectancy, but they also gave the U.S. the 9-11 attacks (or, to be more precise, contributed to causing the organization/movement which planned and executed those attacks). Troop deployments should have a compelling strategic rationale, which, as I've argued here before, many U.S. troop deployments lack. (2) Putting U.S. soldiers in a country is probably not an efficient development strategy in that there must be more cost-effective, direct ways to achieve development goals, not to mention ways that encourage more local participation and 'empowerment'.
So this research is interesting but I doubt it's going to change any minds about the merits of U.S. troop deployments abroad. If you favored the current U.S. footprint before this article, you are still going to favor it, with maybe an additional small arrow in your argumentative quiver. If you favored reducing that footprint, as I do, your position will likely be unchanged.
P.s. Besides correlation/causation, which can always be raised, there may of course be technical issues with the analysis. But I would have to read the article more closely, and in any case the question is largely beyond my competence.