The resistor is used so that the relays turn off faster than if a snubber diode is used. If the resistor has 10 times the resistance of the coil, then the voltage surge is limited to 120 V so a suitable transistor can be chosen. The current will stop very quickly. With a diode, the current could carry on of a long time compared to the fast flash rate of a flasher if there is a blown bulb.
The 100 uF capacitor charges and discharges through the 2k resistor. It charges when Q1 is on, so the negative end is low. The relay is on, so the top of the 2k is high. When the Q1 turns off, the negative end goes high, and the relay turns off so the flasher bulbs pull the top of the 2k resistor low, and that discharges the 100 uF cap.