Bidirectional Current Source
This technique is called "rail sensing" or "supply sensing" and the 1K resistors that are degenerating the rails are there to measure the supply current, which is dominated by the op-amps output stage.  The signal generated by the these resistors drives the main out board transistors that provide the current boost.  The transistors are within the feedback loop of the amplifier so the non-linearities are partially compensated for.  

The large capacitor is to decouple the new "mushy" rails voltage and to provide charge storage while the op-amp is slewing. typical stabilization caps are on the order of 10'2 - 100's of pF's this is 100X to 1000X too large for that purpose.

To understand the operation you need to understand how the device is made.  The feedback is not shown (in the app note schematic) so how the device works in this configuration is not clear. i.e. "how do you get the outside transistors inside the feedback loop?"  Here is the snip from the datasheet.



The op-amp rails respond more to changes in the load than it would if it were just driving the load by itself.  The load comes on, the op-amp slews to make up for it, this increased current is detected and drives the out-board transistors which then shoulder the burden.  As the output approaches the final value the op-amp rail current drops. The decoupling capacitor (10 uF) need only supply the charge for slewing of the op-amp.