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.