understanding the spaceship interface circuit logic

I started with Arduino two days back. I attempted the spaceship interface and it worked perfectly. However, I was not able to understand the circuit and hence seeking help for that. Specifically, we have a 10K ohm resistor that is connected to the switch button and the negative row of the breadboard.

This resistor brings the system back in the initial condition (green on, both red off) once the switch is released. So, if we remove this resistor, and the button is pressed and released, the lights keep blinking. The system does not go back to the initial condition.

How does this resistor bring the system back in the initial condition?

CMOS chips (basically all modern logic chips) have extremely high resistance inputs (10 gigaohms or more),
so if they are unconnected the voltage on them can floats around, and is undefined (ie vary at random, pick
up nearby electrical noise or such).

A simple push-switch either makes contact or not. If not then the pin is free to float unless some other
device defines its voltage, such as a resistor to the other voltage rail from the switch. Switch open, the
resistor defines the voltage, switch closed the switch (being very low resistance) wins from the resistor
and defines the voltage at the other rail.

How does this resistor bring the system back in the initial condition?

It doesn't, it is a pull down resistor for the input to keep the pin's voltage at zero when the button is not pressed. It stops the pin floating and picking up random crap.
See http://www.thebox.myzen.co.uk/Tutorial/Inputs.html