I had a nice reply outlining how it works and then Firefox froze up on me.
I'll break it down into the main blocks:
The footswitch is connected to an RC network that slows down the high-low transition and the 555 goes high when the input goes below 1/3 of the supply. The 555 also has hysteresis so altogether this is a debounce circuit. Nothing too fancy here.
The output of the 555 goes high and turns on S4, which turns on S1. S2 and S3 briefly see a logic high due to the rising edge that is let through by the 100n cap, but then they see a logic low again when the charge is bled to ground by the 1k resistor. So S2 and S3 close briefly at the beginning of the process.
S1 stays on as long as the 555 output stays high, so you keep holding the button down and current flows through the 100k into the op-amp. The 100n cap across the op-amp makes it an integrator, so the current flowing in causes the output voltage to ramp up slowly.
After a while you release the footswitch. This transition is debounced and the 555 output goes low. S4 opens, S2 and S3 remain off (no change) and the control input for S1 goes low due to the 12k and pot resistance to ground.
This open-circuit at the input of the op-amp means no current flows and the output voltage is held at the current value.
If you were to press the button again the op-amp output would again go low and slowly ramp up, release the button again and it will hold at a new value.
Now we can look at the discrete buffer. The 0-5V output of the integrator is divided in half by the 10k:10k resistors and the resultant 0-2.5V is buffered by the BJT op-amp. The output is connected to pin 6 of the PT2399. The PT2399 will respond to a control voltage between 0-2.5V by varying its delay time from minimum to maximum, respectively.
So finally the behavior of the circuit as a whole is this:
Press and hold footswitch for X milliseconds, then release. This causes the PT2399 delay time to first reset to min, then ramp up to Y milliseconds and finally remain at Y milliseconds after the footswitch is released.
The slope trim potentiometer allows you to adjust the ramp speed such that X=Y. Under this condition the footswitch is a delay time input. Hold the switch closed for 300ms, get a 300ms delay.
[edit]
Here is an updated circuit which includes the power-on reset to prevent latch-up, and the integrating resistor value is changed to 1M. The range of adjustment is between 300ms and 2000ms full-scale. That means it will take somewhere between .3 and 2.0 seconds to get the PT to produce its maximum delay time. This ought to cover any variation between different PT chips.
