Very simply, the snubber provides a load to dissipate energy from rapid changes in voltage associated with reactive components. In your diagram, consider what happens when the triac experiences a voltage spike. The triac itself has capacitance between the line and the gate, and the spike may provide enough current through this capacitance to turn it "on."
The capacitor provides a high current path through the resistor to ground, preventing the spike voltage from rising quickly and providing a resistive load to dissipate the energy. At lower frequencies (like the motor frequency) the capacitor impedance is high and the current is low.
A triac is a form of thyristor, so it can't turn off when current is flowing. And if the triac never switches off while current is flowing through the inductor, the snubber is completely useless.
But of course, the world we live in is not ideal and the triac will turn off before the current has completely stopped, so my guess is that the snubber forms a path for the current that is too low to go correctly through the triac.
If you're using a device that can turn off (eg. MOSFETs or gate-turn-off thyristors), then the snubber deals with the full load.