Why is there a divider circuit here?
That's a pull-down resistor for the Arduino's input pin. It's there to handle the case when the button is not pushed. Without the resistor, when the button is not pushed, nothing would be connected to the input pin. The pin would have an undefined voltage, which the Arduino might interpret as either a logical HIGH or LOW. Your program that needs to read the button state would not work correctly.

If you tried to replace the pull-down resistor with a straight piece of wire, it would create a short-circuit between +5V and ground whenever the button is pushed. So a resistor is required.

There's nothing magic about the value 10KΩ, but it's a common "standard" resistor value for pull-downs and pull-ups in digital circuits. The resistor must be large enough to prevent dangerous levels of current from flowing between +5V and ground when the button is pushed, but also small enough that the input pin is correctly sensed as a logic LOW when the button is not pushed.