OK, let's try to keep it simple.
Two facts:
1. A transistor (NPN type) will not turn on unless the base voltage is greater than about 0.6 V
2. The voltage difference across a capacitor can only change when current flows into it.
So, suppose transistor Q1 is turned off. Since only one transistor can be on at a time (you figure out why), then Q2 is turned on.
Now, current is flowing through the left 10k resistor and trying to turn Q1 on, but it can't because the current is being absorbed by the capacitor also attached to Q1 base. The current flowing into this capacitor is raising the voltage difference, but only slowly, which is why the circuit can only flip-flop slowly from one state to the other.
Eventually, the voltage at Q1 base will increase above 0.6 V and will cause Q1 to turn on. As soon as Q1 turns on the voltage on Q1 collector will drop close to zero. Remembering fact 2, the voltage on Q2 base will also drop rapidly, turning Q2 off (remember it was on before).
When Q2 turns off its collector voltage will rise, causing the voltage on Q1 base to rise (fact 2 again). Therefore Q1 will rapidly turn on, causing Q2 to be firmly turned off.
The circuit is now flipped, and current is flowing through the right 10k resistor, trying to turn Q2 on. Eventually the voltage at Q2 base will rise above 0.6 V and the circuit will flop back again.
Everything is now back where we started and the cycle repeats.