Looking at the transistor base voltages will help you see how it works.
Assume that Q2 is on and Q1 is about to turn on.
When Q1 turns on its collector goes from 5V to 0V.
This is coupled through the C1 to Q2's base, which had been turned from current through R3 on with about 0.7V Vbe.
The 5V-0V through C1 causes Q2's base to go to about -4.3V, turning it off.
(This generates a 0-5V pulse at Q2's collector which generates a positive pulse through C2 and into Q1's base, fully turning it on.)
C1 now charges through the 5kΩ resistor until it reaches 0.7V which again turns on Q2, which in turn turns Q1 off, to complete the half cycle.
A half cycle takes about 2/3 of an R2C1 time-constant.