This chart makes no distinction between those who've left the labor force because they've given up the hunt for work, and those who've left the labor force because they don't actually want a job (i.e. retirees, and people who were only working so they could get health insurance).

Contrary to what you read on the internet, the BLS actually does track labor force dropouts. There are even multiple categories of them, which you can find in BLS Table A-16: Persons not in the labor force. The data goes back as far as 1994, and graphs are available.

If you look at the numbers, you'll see that the number of people outside the labor force is on a pretty steady upward trend, as it has been for decades. That's just a consequence of population growth. The number of people outside the labor force who actually want a job is also on an upward trend, but a much shallower one. And the people who have specifically given up looking for work, the people whose only reason for not looking for a job was "there aren't any jobs out there"? That's on a downward trend, from an average of 1.17 million in 2010 to 739k in 2014.

739k discouraged workers affects the unemployment rate, but not as significantly as this article makes it seem. In fact, that's something the BLS calculates, as well! It's the U-4 unemployment rate, which is currently at 6.0% (down from 6.1% last month). You can find all unemployment rates, U1 through U6, in BLS Table A-15: Alternative Measures of Labor Underutilization.

Also, the article makes a pretty misleading claim:

After holding steady at roughly 66 percent from 2004 through late 2008, the labor force participation has been falling, and falling, and falling some more, with no end in sight.

The national labor participation rate peaked in 2000 and has been falling ever since. The decline slowed during '04-'08 due to the economic bubble, then steepened when the bubble popped. Since the recession, it has simply returned to its pre-bubble downward trendline.