You cannot change the flashing rate of the LEDs by just connecting the LEDs in series to the main. When you do so, the LEDs will turn ON/OFF 50 or 60 times every second - most probably, you will not see it flashing.
The box you marked in red contains a controller that changes the flashing frequency every time you press the button. This has to be done using a microcontroller, or some sort of PWM, Something that controls the rate of switching the LEDs ON/OFF.