Why are capacitors needed in dimmer circuits?
Because what you need to do is delay the turn-on point, and a capacitor together with a resistor form a delay, characterised by the RC time constant.
The delay should be adjustable anywhere between 0 degrees (full brightness) and as close as you can get to 180 degrees (minimum brightness), to cut out almost all of each half cycle.
The suggested approach of using a potential divider, with no capacitor, will allow delays up to 90 degrees (the peak of the AC waveform) and no further. Which doesn't give an adequate dimming range for most applications.
Resistive divider will also be quite sensitive to noise and spikes on the rising edge of the AC waveform, which doesn't always resemble a nice clean sinewave. Uneven dimming and random flickering is likely to result, so you'll realistically need a capacitor to suppress noise anyway.