Here's the link to the circuit:
http://rustamaji.net/resource/images/flip-flop2.swf
From the simulation, when the 22k resistors are removed the LEDs stop blinking because the current flowing only around capacitor. When either the other resistors are removed or all resistors are removed, the components on the circuit burn out.
What is the function of resistors in this flip-flop circuit?
The 22K resistors supply base current to turn the transistors on. Whichever transistor turns on faster, pulls its collector low, and that falling voltage is coupled by the capacitor to the other base, keeping the other transistor turned off.
Eventually that capacitor charges up and base current starts to flow into the second transistor, which starts turning on. Now its capacitor starts stealing the first transistor's base current turning it off.
So the 22k resistors have 2 functions :
(1) they supply base current to the transistors, and
(2) they - and the capacitors - determine the time delay between transistors switching on, and thus the frequency of oscillation. If they are unequal (or the capacitors are unequal) the duty cycle is not 50% because "on" and "off" times are different.
The 470 Ohm resistors are meant to protect the LEDs from high currents: you can find more explanation here.
The 22k resistors are used together with the capacitors in order to create a time constant that allows the circuit to oscillate at a given frequency. The circuit works as an astable multivibrator: you can find examples here and here.
