What purpose does the capacitor serve in this schematic?
Likely an error
The overall quality of Arduino connected hardware designs varies widely as the community consists of a comparatively large number of hobbyists and students.
The 0.1uF cap was most likely intended to decouple the adjacent +5V power supply line. Attaching it to the latch signal line serves no unique beneficial purpose and can cause problems if your firmware operates the clock line at its maximum frequency and you have a long serial chain of registers.
Delay
If you need to delay the latch pulse to comply with setup and hold times (timing minimums) of the 74HC595 you are using, then you should address this in other ways: either correct your layout and/or add the delay in your firmware.
Debounce
On its necessity...
jippie said it best:

The type of bouncing where this configuration works, is not a digital controller output like the your diagram implies. It is more intended for a push button.

On its consequences...
Even if we believe that a bouncing source was driving this line, there is no practical consequence of such behavior.
Wouter van Ooijen, said it best:

Debouncing the (data) clock input might serve some purpose. Debouncing the latch (clock) input is nonsense: multiple pulses on this input will simply re-load the same data from the shift register to the holding register (latch). -- Wouter van Ooijen