Some of this might be due to a difference in question wording, but that can't account for all of it. It's too big a shift. The obvious conclusion is that public support has dwindled thanks to Rand Paul's filibuster and related questions over John Brennan's nomination to head the CIA.
If that's the case, then you'd expect support to have dropped much more dramatically among Republicans than Democrats. Unfortunately, no internals are available for last year's poll, so we can't tell. Maybe I should ask the Washington Post's polling director if they can make those numbers available. I'd be curious to see how this has really played out.
UPDATE: I misread the Post poll from a year ago. The net result is that 65 percent of the public approved of drone strikes on American citizens, not 79 percent. I've corrected the text. I'll post another update if I get hold of poll internals showing how opinion has shifted among Democrats and Republicans.
UPDATE 2: Peyton Craighill of the Post has kindly sent along the internals of last year's poll. Here's how the net approval for drone strikes against American citizens has changed:
2012 Today Net Change Democrats 58% 41% -17 Republicans 76% 50% -26 Independents 65% 35% -30
There's less of a difference here than I would have guessed. Republican support did indeed drop more than Democratic support, but not by a huge amount. And Independent support dropped by more than either.
At a guess, I'd say this suggests that maybe half the drop is based on a genuine reduction in support over the past year for drone strikes on U.S. citizens, while the other half is a semi-partisan reaction to Rand Paul. But that's just a wild guess.