The so-called Great Moderation occurred in the industrialized world, not just the U.S. Here is the evidence:
As Ben Bernanke has pointed out, the volatility of real GDP was lower before the 1970s and after the 1970s. Here is Bernanke’s article:
So the explanation for the Great Moderation has to do with the breakup of Bretton-Woods. Before 1971, there was an international monetary system that to a certain degree coordinated world-wide inflation with the dollar. When the Fed inflated if gave other countries room to inflate their currencies as dollars wound up in those countries through foreign exchange. Part of the impact of Fed monetary inflation was felt overseas and, if coordinated with other countries, resulted in less dramatic booms and busts.
Bretton-Woods began to unravel in the late 1960s. When it collapsed in 1971, countries allowed their currencies to float (albeit a managed float). When the Fed inflated, the full impact was felt in the U.S. and thus, more volatility in the economy.
In the 1980s, the U.S. put a dollar reserve standard, with pegged exchange rates, back in place. When the Fed inflated the dollar, more dollars were held overseas through foreign exchange and became reserve for the expansion of domestic currencies. This smoothed out the business cycle in industrialized world, but made it more volatile in the less industrialized world. Places like southeast Asia, in particular. After the collapse in southeast Asia beginning in 1997, the effect of Fed inflation was felt more in the U.S. and we had the DotCom bubble followed by the housing bubble.