Current can flow through a capacitor as long as the voltage is changing with time.
An important factor in understanding how that circuit works is knowing that a capacitor works like a straight piece of wire when you consider short time scales. This means that if you change the voltage on one side of a capacitor, the voltage on the other side will immediately change by exactly the same amount (just the same as a wire--if you change the voltage on one end of a wire the voltage on the other end will follow it).
So for example, if Q1 turns on it will pull the + side of C1 low. This will immediately pull the other side of C1 low at the base of Q2 and turn off Q2. If Q2 turns off it will cause the + side of C2 to go high, which will raise the base of Q1 high and keep it turned on. Therefore you have a state with Q1 turned on and Q2 turned off.
Starting from this position, consider each of the capacitors. Will each one charge or discharge, and what will happen when one or other of them does so?