What's the role of the capacitors in this AC circuit?
background

Capacitors will always conduct current if there is a change in voltage across it.

Ic=CdV/dt  However the current is not determined by the physical size as the density of nF,uF is determined by the dielectric constant and electrolytics and batteries alike are much greater than ceramic and plastic caps since they are highly corrosive (acidic) and thus can stored more charges.

Here you will likely find they are 400 or 600 V rated plastic caps that may be able to handle > 1kV transients.  

answer

Caps have many applications in time and frequency sensitive circuits.


resonate with inductors to perform LPF, HPF or BPF depending on values and circuit
integrate current with a current-sense resistor to convert "int(I.in) to V.out"
differentiate voltage to create a current then sense this to convert "dV/dt to V.out"


Here


C1 is a LPF to attenuate noise > 100kHz from switching current &lt; 1us an also absorb some external noise
C2 C3 are LPF time/phase delay circuits cascaded with variable RC=T values
C3 acts like a tiny battery each cycle that activates the DIAC switch which in turn discharges C3 as it's voltage then triggers the Triac. 
Then the Triac latches ON and no longer needs the voltage from C3. 
This repeats every half cycle  as the Triac turns OFF (unlatched) when current drops near the zero crossing.