Silly me, it does use a regulated supply so ignore my comment on regulation.
The 555 is triggered every time the AC signal passes through 0V.
The control voltage sets the time delay before the TRIAC is fired. The higher the control voltage, the later the TRIAC will fire, causing less power to be sent to the load.
The calculations for Ct and Rt can be found on the datasheet for the 555 timer and will depend on the control voltage.
The comparator performs the zero crossing function, it can be replaced with a transistor to save money but it won't work so well because the trigger voltage will be 0.6V rather than near 0V.