There are two separate points of interest you're looking at so I'll separate this into sections.
Sudden Drop at Day 1559
As near as I can tell, this is the result of a quarterly roll of the satellite, specifically the end of quarter 17. Every 90 days, which NASA calls a quarter, the space craft does a 90 degree roll to optimize the solar panel efficiency. What this means for the data is that when this roll occurs at the end of the quarter, the photons from the star now fall on a different CCD channel which is calibrated slightly differently, causing a slight discontinuity in the data. You can read about "Quarter Stitching" here. I've extracted some relevant text and an image for posterity.
After each roll maneuver, most Kepler targets fall on a different CCD channel and the target’s point-spread function will be distributed differently across neighboring pixels. Naturally this redistribution requires a new computation of the target mask and optimal aperture size within the Kepler pipeline, taking into account the point-spread function, new CCD characteristics, and new estimates of nearby source crowding. The operational outcome is often a different mask shape, with differing amounts of flux within the optimal aperture from both the target and contaminating sources.
Light curves extracted from all single pixels within the quarter 4, 5 and 6 Target Pixel Files for KIC 9603833 (the symbiotic star StHA 169). Gray pixels comprise the optimal apertures that yield the archived light curves. The target point-spread function is distributed across neighboring pixels differently from quarter to quarter and hence the optimal aperture varies in size and shape from quarter to quarter.
Flat line between days 1581 to 1587
I'm less certain about this one. I believe the flat line is just missing data. To keep the line continuous, they just connected the data points they had with a straight line.
Why the data is missing, I can't be sure. I know that Quarter 17 ended with the failure of Kepler's second reaction wheel so my guess is that this missing data is in some way related to these reaction wheel troubles. I do know they tried,in the past, to put the spacecraft in 10-day "wheel rest" mode in hopes that it would give the reaction wheels a break, let them get lubed up again, and be better when they restarted. Perhaps, by this stage, they had already suspected a second reaction wheel problem and were trying out a wheel rest mode. I can't find any evidence a wheel rest occurred during Quarter 17 though, so this is just my guess.