Square wave generation happens at the trigger terminal (2nd PIN) of 555 IC. When the power supply is turned ON then at the very beginning the capacitor starts storing the charge and so the potential across the capacitor increases so to trigger the IC and pushing it into ON. After a certain time the capacitor gets enough potential to discharge through R2. At this point the IC output is turned to OFF state and it stays at OFF until capacitor starts charging again. And so we have square wave at the output.
Now the square wave output of 555 IC is fed into counter as clock, so every time a peak is passed the counter considers it as an event and increments the output by one for every passing event. Once it reaches its limit to track the events, it automatically resets to zero and starts again to count the pulses. And for the output it provides the event number as binary output through pins 9,7,6,5,3,2,4,13,12,14,15,1. In LSB to MSB manner. So if the event count is 10 the pins (7(2^1=2),5(2^3=8)) will be high and so the corresponding LEDS glow. To reset the counter to zero under any stage connect the MR pin of counter to +5V, this resets the counter to zero.
