How does an astable multivibrator LED blinking circuit work?

This circuit is called an astable multivibrator, and the reason this circuit works is a little difficult to put into words.

Take a look at this circuit simulation that shows visually what is happening. You can slow the speed right down and look carefully at how the current is flowing.

While one of the capacitors is charging, current flows to the base of the alternate transistor, making the emitter-collector path conduct, making one of the LEDs light. When the capacitor is charged, it stops conducting and switches off the transistor, and then the other capacitor begins to charge switching on the other transistor, at the same time the first capacitor discharges, then the cycle repeats.

Imagine that R2 were omitted, and suppose the left side transistor (which I'll call Q1) started out being turned on. Then the right side transistor (Q2) would have nothing to turn it on, and the left side transistor would be held on by current through R3, while the R4 and the LED would charge C2. The effect would be that the left LED would come on and stay on, while the right LED would never light.

Now add R2. This will cause C1 to charge to -0.7 volts, until Q2 turns on. Once that happens, the charge on C2 would cause the base of Q1 to go negative, turning it off. When that happens, left-side LED current will start flowing through C1 and the base of Q2, turning it on even harder. Once Q2 has turned on, it will keep Q1 off until C2 charges to -0.7 volts.

Note that while the circuit would have a stable state with both transistors on and both capacitors reverse-biased by 0.7 volts, in practice the winky-blink circuit always starts oscillating rather than entering such a state.