What happens when you rectify phase cut current?
If the cap is big enough, then its voltage will be mostly the peak of the input voltage with a little ripple.

Simplistic dimmers modulate the RMS output by keeping the output off a variable delay after each zero crossing.  You can think of the full output as having 0 delay, and full off as having &frac12; cycle delay (8.3 ms for 60 Hz).

Since the capacitor voltage rides the peaks, there will be little change in this voltage during the first half of the delay range.  Since the peak voltage occurs half way thru the possible delay range, the whole first half of the delay range will have the same peak voltage, even though the RMS value coming out of the dimmer is half at half way thru the delay interval.  In the second half of the delay interval, the capacitor voltage will drop since the peak voltage coming out of the dimmer also drops.

These kinds of dimmers are rather frowned upon today due to the high frequency components they create in the current.  They were also intended for loads like incandescent bulbs that respond to the RMS voltage, not the peak.  They are poorly matched to modern electronic loads.