It's a pull down resistor, its purpose is to set the voltage of the Arduino input to 0 V when the button is open. It just set a default voltage value to the connected node.
But, why such a resistor in needed? Well, when the button is open, the voltage is not totally equals to zero due to antenna effect or surrounding component. It is then possible that the Arduino input is trigger for no reason.
Note that the value of pull up or pull down resistors use to be around 10k - 100k. If the value is to low, it's going to draw to much current when the button is close. If the value is too high, the resistor will behave as an open circuit, making the pull down resistor useless.
As it is widely used in electronics, I recommend you to read some more about this concept.