Okay, they only consider the latter a mystery, but for those who follow the data both are equally mysterious. The piece was titled "an economic mystery: why are men leaving the workforce?" The piece noted the reduction in the percentage of prime-age men in the workforce from nearly 100 percent in the 1960s to 88.3 percent at present. It then said that no one really knows why there has been this decline.
Actually, it really is not much of a mystery. While the piece wants to attribute it to the peculiar situation men face in the labor market, it is worth noting that there has also been a sharp decline in the percentage of prime-age women in the labor market. (Actually, a better measure is simply looking at the share of people who are employed. Many workers stop saying they are looking for jobs when they are no longer eligible for unemployment benefits. With a sharp reduction in eligibility for benefits over the last three decades, people who are not working are now much less likely to say they are looking for work.)
The figure below shows the percentage of prime-age women that are working since 1990.
The chart shows that after rising sharply from 1993 to 2000. It then fell sharply following the 2001 recession and again in 2007–2009 recession. It has since risen in the recovery but it is still 3.8 percentage points below the peak hit in 2000. The pattern among prime-age men is similar, although the employment rate is now 4.8 percentage points below the 2000 peak. (Remember the EPOP for women had been rising before the 2001 recession and was projected at the time to continue to rise.)
The fact the EPOP for men and women has followed a very similar pattern since 2000, and the declines have been associated with weak demand in the economy, suggests that the explanation might be weak demand. In other words, the problem is not too many men getting disability or lacking the skills needed in a 21st century economy, but rather just not enough demand.
If we could get the trade deficit down or got the government to spend more money then we could boost demand and get the prime-age men discussed in this piece back to work. We would do the same for prime-age women. (We could also follow the German model and reduce the length of the average work year, thereby spreading around available work.)
So the villains are not lazy men, but rather folks like Paul Ryan, Peter Peterson, and the Washington Post who insist that we have to keep our budget deficits low even in a context of near record low interest rates and very low inflation.
It is also worth mentioning that the lazy welfare cheat story doesn't fit the data at all. Benefits of various stripes have gotten more stingy over the period when men's EPOPs were declining. Also, as the report cited in the piece from the President's Council of Economic Advisers notes, the United States ranks near the bottom among wealthy countries in the generosity of benefits. It also ranks near the bottom in prime age EPOPs.