How does this TLC556 work?
The capacitor C2 (with R3) AC-couples the trigger signal so that it generates a ~1ms pulse on every negative edge of the first timer. The time constant of R3 C2 is more like 100usec so it does not hold the 1ms pulse low, and recovers in plenty of time for the next edge. 

You might think that a similar effect could be created by shunting R2+P1 with a diode and series resistor (and doubling C1) but the resistor would have to be unpleasantly low (in the 10 ohm range) so the currents would be limited mostly by the IC output.  

The pulses measured at the trigger input will also be affected by the ESD protection network on pin 8. A prudent designer might add a resistor in  series with C2 to limit the protection diode input current on the rising edge of pin 5 to a mA or so. Fortunately the source capability of the TLC chip is rather limited so it doesn't destroy itself, but it's operating outside of abs-max specs (which are difficult to meet - maximum input voltage is stated to be Vcc and that gets exceeded every time pin 5 goes high after being low).