In the picture shown the switch is open. In an ideal environment the voltage at "1" will be 0. Thus, it will have the same potential as the ground.
If we are in a noisy environment, we have to imagine that there is a small capacitance in parallel with the resistor to ground. This is called a
parasitic capacitance, since it is an unwanted but always present nuisance.
When this capacitor is there, and we have incoming noise, the voltage may rise. Let's say the voltage suddenly rises to 1 volt because of incoming electromagnetic interference. Now there is a voltage across the resistor to ground, and current will flow. As the current flows, the capacitor will be discharged and return to 0 volts. The rate at which this happens depends on the capacitor value (very very small) and the resistor value. If the resistor value is very very high, then there will only be a small current flowing, and the potential will decrease more slowly, than if the resistor was small.
That is the first part of it. Now, the voltage rise (from the noise) is actually dependent on the resistor value.
If we imagine a new wire connecting to point "1". This wire is a "virtual" one, and is only there to signify the incoming noise. The noise is modeled as an incoming current that lasts for let's say 2 microseconds.
Now, if the resistor is small, more current will flow through the resistor, and the voltage at "1" after the incoming current is over is less than if the resistor had been big.
The conclusion of this is that a small resistor will help you against noise, but it will also increase power consumption.