Traditionally, macroeconomists have believed that low interest rates encourage inflation. But first Japan, and now the U.S. and Europe have kept rates low for years now, and inflation has stayed stubbornly low. A radical group of macroeconomists, including Stephen Williamson of the Federal Reserve Bank of St. Louis and John Cochrane of the Hoover Institution, have introduced a new theory called Neo-Fisherism, which says that a long period of low interest rates actually holds prices down instead of pushing them up. Williamson and Cochrane have both repeatedly stressed that New Keynesian models -- the most mainstream type of macroeconomic theory -- can easily yield the Neo-Fisherian result instead of the traditional view.
The first sentence is flat out wrong. There is nothing in traditional macroeconomics that suggests low interest rates lead to high inflation. Paul Krugman would define traditional macro as the IS-LM model, so let's use that workhorse. If the LM curve shifts right, then interest rates fall and inflation rises. If the IS curve shifts left, then interest rates fall and inflation falls. So there is no necessary relationship between interest rates and inflation. If Japan has experienced low rates and low inflation, presumably the Japanese IS curve has been sluggish---no big surprise given their falling population. So contra Smith, there is no "puzzle" to be explained.
As far as Neo-Fisherism, we've known for a long time that low nominal interest rates tend to be correlated with low inflation. Milton Friedman said something similar back in 1997.
Friedman is saying that a tight money policy will cause low inflation, and that over time this will lead to low interest rates. So if you observe low interest rates, it's a sign that policy was tight in previous periods. Friedman understood all this, and hence there is no puzzle for the Neo-Fisherians to explain. If you want to claim that a sudden unexpected cut in interest rates will lead to lower inflation expectations, that's different. There are rare occasions where it is true (such as Switzerland in January 2015) but only when accompanied by other contractionary steps. Normally an unexpected rate cut leads to higher expected inflation (in the asset markets).
Smith goes on to discuss how new ideas in behavioral economics may be able to resolve some of these interest rate puzzles. I'm all for trying new approaches, including behavioral economics (indeed my other blog is named after a popular behavioral economic theory) but in this case I simply don't see any puzzles to explain. Rates are low because tight money has led to very slow NGDP growth, plus a few other non-monetary factors impacting global saving/investment.