Reverse Engineering of a AC speed controller
The RC pair \$C_2 \cdot (R_4 + R_1||R_2)\$ provide a delay as C2 charges from the initial voltage to the breakdown voltage of the diac (typically around +/-35V). It's not calculated as a simple RC delay since the voltage is a half sine wave.  

When that breakdown voltage is met, C2 discharges through the triac gate, triggering it and the remainder of the AC half-cycle passes through to the output.

C1+R3 form a snubber for the triac. 

The trimpot is used to adjust the low end of the adjustment, as pots (and capacitors) have poor tolerance. 

As others have said, this is just a old-style triac light dimmer circuit (minus any attempt to mitigate EMI) such as you might have found in a house 30 or 40 years ago.