tl;dr: The September 15, 2017 end for Cassini is part of the second mission extension (XXM, or Solstice Mission) proposed in 2009. The plan was approved.
above: "Choosing the XXM Tour: Cassini project manager Bob Mitchell demonstrates the features of the Cassini spacecraft to the XXM senior review panel at JPL on February 10th 2009, watched by project scientist Bob Pappalardo (left), lead mission planner David Seal (right), and members of the review panel." Credit: John Spencer. From the 2009 blog post Cassini's Proposed Extended-Extended Mission Tour.
This excellent, concise answer to the question What force is bringing Cassini down into Saturn's atmosphere in another 145 days; drag, or…? describes how the whole thing is going down, both literally and idiomatically.
Cassini's August 8 gravitational encounter with Titan sets it up for its September 11 encounter with Titan, which then sets it up for its deep-dive into Saturn's atmosphere on September 15.
This is not like a de-orbit burn you might be imagining, it is a pair of highly choreographed gravitational assists from Titan flybys, lowering the periapsis enough to assure annihilation.
The Cassini mission has been incredibly successful in terms of both observation of Saturn and many of its moons at close range, and in a demonstration of orbital acrobatics.
Because of this amazing success, the Cassini mission has been extended twice already, at significant cost, in 2008 and again in 2010. See below.
The internet search term would be Cassini Ball of Yarn; see also Unravelling Cassini's “ball of yarn” orbit around Saturn, tabulation of propulsive maneuvers?.
Full video here.
More insight can be found in the 2009 John Spencer blog on The Planetary Society's website Cassini's Proposed Extended-Extended Mission Tour. The double encounters with Titan and burn up in Saturn's atmosphere on September 15, 2017 had already been planned and proposed and waiting for approval back in 2009:
It seems like no time since we selected Cassini's extended mission tour of the Saturn system, in early 2007. Now we're flying that tour, which extends Cassini's original four years in Saturn orbit for another 27 months, until September 2010. So now we're looking into the future- far into the future. If NASA approves the mission plan that was recommended by the Cassini project in late January, and if the spacecraft remains healthy, we hope to continue Cassini's exploration of Saturn for another seven years, until 2017. The ungainly working title of this planned new mission phase is the extended-extended mission, or XXM.
The year 2017 is special because May 2017 marks northern midsummer in the Saturn system, and the more elegant alternative name for the XXM is the Solstice Mission. A Saturn year is 29.4 Earth years, and the considerable 27 degree tilt of Saturn's pole (which, incidentally and bizarrely, can probably be blamed on the planet Neptune), means that the slow progression of the seasons has major effects, especially on Saturn itself and on its giant moon Titan.
The blog is extensive and informative and definitely worth a read. It further explains that the final orbits of Cassini started back in April of 2017. The difference between April 22 and September 15 2017 is 146 days, and the basis of the title of the question What force is bringing Cassini down into Saturn's atmosphere in another 145 days; drag, or…?:
Cassini will execute 20 of these close "F-ring" orbits before setting up for a final close Titan flyby on April 22, 2017. This flyby will do something astonishing: it will perturb the orbit so that Saturn closest approach jumps, in a single leap, from just outside the main ring system into the narrow zone of safety between the inner edge of the innermost ring (the D ring) and the planet itself, just 3,800 kilometers above Saturn's cloud tops. Cassini will continue to thread this needle for 23 orbits (called, with some understatement, the "proximal" orbits) until a final distant nudge from Titan on September 11, 2017 delivers the death blow, altering the orbit just enough to drop Cassini into Saturn on September 15.
Later in the article:
There remains one major hurdle before we can put these plans in motion: money. Cassini will need another seven years of funding, and NASA has many other claims on its budget. To sweeten the deal the Cassini project has found ways to simplify operations by concentrating efforts on the highest-priority science, and it looks like we can run each year of the XXM on considerably less money than we need to maintain the more hectic pace of the prime and extended missions. A couple of weeks ago I participated in a "Senior Review" at JPL, where we made the scientific case for the XXM to an external panel of scientists. The panel will pass on their recommendations to NASA headquarters, and we will wait more-or-less patiently for the final decision. In the meantime, we have lots of preparatory work to do to flesh out the details of the XXM. If and when we get the go-ahead from headquarters, we will be ready.
And for balance, from Wikipedia:
On April 15, 2008, Cassini received funding for a 27-month extended mission. It consisted of 60 more orbits of Saturn, with 21 more close Titan flybys, seven of Enceladus, six of Mimas, eight of Tethys, and one targeted flyby each of Dione, Rhea, and Helene. The extended mission began on July 1, 2008, and was renamed the Cassini Equinox Mission as the mission coincided with Saturn's equinox.
Second mission extension
A proposal was submitted to NASA for a second mission extension (Sept 2010 – May 2017), provisionally named the extended-extended mission or XXM. This ($60M pa) was approved in Feb 2010 and renamed the Cassini Solstice Mission. It includes Cassini orbiting Saturn 155 more times, conducting 54 additional flybys of Titan and 11 more of Enceladus.